Interactive Algebra & Trigonometry - A Self Guided Study Companion

home *** CD-ROM | disk | FTP | other *** search
/ Interactive Algebra & Tri…f Guided Study Companion / Interactive Algebra and Trigonometry - A Self-Guided Study Companion.iso / tutor / chap_2 / 2-4-3.tut < prev next >
Unknown | 1996-07-18 | 13.9 KB
open in:
MacOS 8.1 |
Win98 |
DOS
view JSON data |
view as text

This file was not able to be converted.
This format is not currently supported by dexvert.
Confidence	Program	Detection	Match Type	Support
`1%`	dexvert	Eclipse Tutorial `(other/eclipseTutorial)`	ext	Unsupported
`1%`	dexvert	JuggleKrazy Tutorial `(other/juggleKrazyTutorial)`	ext	Unsupported
`100%`	file	data	default
`100%`	gt2	Kopftext: 'TUTOR 06'	default (weak)
hex view
+--------+-------------------------+-------------------------+--------+--------+
|00000000| 54 55 54 4f 52 20 30 36 | c4 32 00 00 fc 04 00 00 |TUTOR 06|.2......|
|00000010| 00 17 4f 04 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 54 |..O....W|rong.  T|
|00000020| 68 69 73 20 69 73 20 74 | 68 65 20 61 6e 73 77 65 |his is t|he answe|
|00000030| 72 20 66 6f 72 20 28 11 | 33 66 11 31 2f 11 33 67 |r for (.|3f.1/.3g|
|00000040| 11 31 29 28 2d 31 29 2e | 20 20 49 74 20 61 70 70 |.1)(-1).|  It app|
|00000050| 65 61 72 73 20 74 68 61 | 74 20 79 6f 75 20 66 6f |ears tha|t you fo|
|00000060| 72 67 6f 74 20 74 6f 0d | 0a 00 73 75 62 74 72 61 |rgot to.|..subtra|
|00000070| 63 74 20 11 33 67 11 31 | 28 33 29 2e 0d 0a 00 00 |ct .3g.1|(3).....|
|00000080| 17 4f 03 0d 0b 00 52 69 | 67 68 74 2e 20 20 41 66 |.O....Ri|ght.  Af|
|00000090| 74 65 72 20 65 76 61 6c | 75 61 74 69 6e 67 2c 20 |ter eval|uating, |
|000000a0| 77 65 20 6f 62 74 61 69 | 6e 20 2d 34 2e 0d 0a 00 |we obtai|n -4....|
|000000b0| 00 17 4f 05 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 57 |..O....W|rong.  W|
|000000c0| 65 20 66 69 6e 64 20 28 | 11 33 66 11 31 2f 11 33 |e find (|.3f.1/.3|
|000000d0| 67 11 31 29 28 2d 31 29 | 20 2d 20 11 33 67 11 31 |g.1)(-1)| - .3g.1|
|000000e0| 28 33 29 20 61 73 20 66 | 6f 6c 6c 6f 77 73 2e 20 |(3) as f|ollows. |
|000000f0| 20 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 | .0...  |        |
|00000100| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000110| 20 11 32 32 0d 0b 00 20 | 20 20 20 20 20 11 33 66 | .22... |     .3f|
|00000120| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00000130| 31 32 28 2d 31 29 20 20 | 2d 20 31 20 20 20 20 20 |12(-1)  |- 1     |
|00000140| 20 20 20 20 20 20 20 20 | 32 20 2d 20 31 0d 0b 00 |        |2 - 1...|
|00000150| 20 20 20 20 20 28 11 34 | 32 11 31 29 28 2d 31 29 |     (.4|2.1)(-1)|
|00000160| 20 2d 20 11 33 67 11 31 | 28 33 29 20 3d 20 11 34 | - .3g.1|(3) = .4|
|00000170| 32 32 32 32 32 32 32 32 | 32 32 20 11 31 2d 20 28 |22222222|22 .1- (|
|00000180| 33 20 2b 20 32 29 20 3d | 20 11 34 32 32 32 32 32 |3 + 2) =| .422222|
|00000190| 20 11 31 2d 20 35 20 3d | 20 31 20 2d 20 35 20 3d | .1- 5 =| 1 - 5 =|
|000001a0| 20 2d 34 0d 0b 00 20 20 | 20 20 20 20 11 33 67 20 | -4...  |    .3g |
|000001b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000001c0| 11 31 2d 31 20 2b 20 32 | 20 20 20 20 20 20 20 20 |.1-1 + 2|        |
|000001d0| 20 20 20 20 20 20 20 20 | 20 31 0d 0a 00 00 17 4f |        | 1.....O|
|000001e0| 05 0d 0b 00 57 72 6f 6e | 67 2e 20 20 57 65 20 66 |....Wron|g.  We f|
|000001f0| 69 6e 64 20 28 11 33 66 | 11 31 2f 11 33 67 11 31 |ind (.3f|.1/.3g.1|
|00000200| 29 28 2d 31 29 20 2d 20 | 11 33 67 11 31 28 33 29 |)(-1) - |.3g.1(3)|
|00000210| 20 61 73 20 66 6f 6c 6c | 6f 77 73 2e 20 20 12 30 | as foll|ows.  .0|
|00000220| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00000230| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00000240| 32 0d 0b 00 20 20 20 20 | 20 20 11 33 66 20 20 20 |2...    |  .3f   |
|00000250| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 32 28 |        |    .12(|
|00000260| 2d 31 29 20 20 2d 20 31 | 20 20 20 20 20 20 20 20 |-1)  - 1|        |
|00000270| 20 20 20 20 20 32 20 2d | 20 31 0d 0b 00 20 20 20 |     2 -| 1...   |
|00000280| 20 20 28 11 34 32 11 31 | 29 28 2d 31 29 20 2d 20 |  (.42.1|)(-1) - |
|00000290| 11 33 67 11 31 28 33 29 | 20 3d 20 11 34 32 32 32 |.3g.1(3)| = .4222|
|000002a0| 32 32 32 32 32 32 32 20 | 11 31 2d 20 28 33 20 2b |2222222 |.1- (3 +|
|000002b0| 20 32 29 20 3d 20 11 34 | 32 32 32 32 32 20 11 31 | 2) = .4|22222 .1|
|000002c0| 2d 20 35 20 3d 20 31 20 | 2d 20 35 20 3d 20 2d 34 |- 5 = 1 |- 5 = -4|
|000002d0| 0d 0b 00 20 20 20 20 20 | 20 11 33 67 20 20 20 20 |...     | .3g    |
|000002e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 2d |        |     .1-|
|000002f0| 31 20 2b 20 32 20 20 20 | 20 20 20 20 20 20 20 20 |1 + 2   |        |
|00000300| 20 20 20 20 20 20 31 0d | 0a 00 00 17 4f 03 0d 0b |      1.|....O...|
|00000310| 00 54 68 69 73 20 69 73 | 20 63 6f 72 72 65 63 74 |.This is| correct|
|00000320| 2c 20 62 75 74 20 73 6f | 20 61 72 65 20 28 62 29 |, but so| are (b)|
|00000330| 20 61 6e 64 20 28 63 29 | 2e 20 20 54 68 65 72 65 | and (c)|.  There|
|00000340| 66 6f 72 65 2c 20 74 68 | 65 20 62 65 73 74 20 61 |fore, th|e best a|
|00000350| 6e 73 77 65 72 20 69 73 | 20 28 64 29 2e 0d 0a 00 |nswer is| (d)....|
|00000360| 00 17 4f 03 0d 0b 00 54 | 68 69 73 20 69 73 20 63 |..O....T|his is c|
|00000370| 6f 72 72 65 63 74 2c 20 | 62 75 74 20 73 6f 20 61 |orrect, |but so a|
|00000380| 72 65 20 28 61 29 20 61 | 6e 64 20 28 63 29 2e 20 |re (a) a|nd (c). |
|00000390| 20 54 68 65 72 65 66 6f | 72 65 2c 20 74 68 65 20 | Therefo|re, the |
|000003a0| 62 65 73 74 20 61 6e 73 | 77 65 72 20 69 73 20 28 |best ans|wer is (|
|000003b0| 64 29 2e 0d 0a 00 00 17 | 4f 03 0d 0b 00 54 68 69 |d)......|O....Thi|
|000003c0| 73 20 69 73 20 63 6f 72 | 72 65 63 74 2c 20 62 75 |s is cor|rect, bu|
|000003d0| 74 20 73 6f 20 61 72 65 | 20 28 61 29 20 61 6e 64 |t so are| (a) and|
|000003e0| 20 28 62 29 2e 20 20 54 | 68 65 72 65 66 6f 72 65 | (b).  T|herefore|
|000003f0| 2c 20 74 68 65 20 62 65 | 73 74 20 61 6e 73 77 65 |, the be|st answe|
|00000400| 72 20 69 73 20 28 64 29 | 2e 0d 0a 00 00 17 4f 03 |r is (d)|......O.|
|00000410| 0d 0b 00 52 69 67 68 74 | 2e 20 20 41 6c 6c 20 74 |...Right|.  All t|
|00000420| 68 72 65 65 20 61 6e 73 | 77 65 72 73 20 61 72 65 |hree ans|wers are|
|00000430| 20 63 6f 72 72 65 63 74 | 2e 0d 0a 00 00 17 4f 03 | correct|......O.|
|00000440| 0d 0b 00 52 69 67 68 74 | 2e 20 20 59 6f 75 20 66 |...Right|.  You f|
|00000450| 6f 75 6e 64 20 11 33 66 | 20 11 34 6f 20 11 33 67 |ound .3f| .4o .3g|
|00000460| 20 11 31 61 6e 64 20 11 | 33 67 20 11 34 6f 20 11 | .1and .|3g .4o .|
|00000470| 33 66 20 11 31 63 6f 72 | 72 65 63 74 6c 79 2e 0d |3f .1cor|rectly..|
|00000480| 0a 00 00 17 4f 03 0d 0b | 00 4f 6f 70 73 21 20 20 |....O...|.Oops!  |
|00000490| 59 6f 75 20 6d 75 6c 74 | 69 70 6c 69 65 64 20 11 |You mult|iplied .|
|000004a0| 33 66 20 11 31 61 6e 64 | 20 11 33 67 20 11 31 69 |3f .1and| .3g .1i|
|000004b0| 6e 73 74 65 61 64 20 6f | 66 20 66 6f 72 6d 69 6e |nstead o|f formin|
|000004c0| 67 20 63 6f 6d 70 6f 73 | 69 74 65 20 66 75 6e 63 |g compos|ite func|
|000004d0| 74 69 6f 6e 73 2e 20 20 | 0d 0a 00 52 65 76 69 65 |tions.  |...Revie|
|000004e0| 77 20 47 75 69 64 65 64 | 20 45 78 61 6d 70 6c 65 |w Guided| Example|
|000004f0| 20 34 20 61 6e 64 20 74 | 72 79 20 61 67 61 69 6e | 4 and t|ry again|
|00000500| 2e 0d 0a 00 00 17 4f 03 | 0d 0b 00 4f 6f 70 73 21 |......O.|...Oops!|
|00000510| 20 20 59 6f 75 20 68 61 | 76 65 20 74 68 65 20 63 |  You ha|ve the c|
|00000520| 6f 6d 70 6f 73 69 74 65 | 20 66 75 6e 63 74 69 6f |omposite| functio|
|00000530| 6e 73 20 11 33 66 20 11 | 34 6f 20 11 33 67 20 11 |ns .3f .|4o .3g .|
|00000540| 31 61 6e 64 20 11 33 67 | 20 11 34 6f 20 11 33 66 |1and .3g| .4o .3f|
|00000550| 20 11 31 6d 69 78 65 64 | 20 75 70 2e 0d 0a 00 00 | .1mixed| up.....|
|00000560| 17 4f 06 0d 0b 00 4e 6f | 2e 20 20 59 6f 75 72 20 |.O....No|.  Your |
|00000570| 61 6e 73 77 65 72 20 66 | 6f 72 20 11 33 67 20 11 |answer f|or .3g .|
|00000580| 34 6f 20 11 33 66 20 11 | 31 69 73 20 63 6f 72 72 |4o .3f .|1is corr|
|00000590| 65 63 74 2c 20 62 75 74 | 20 74 68 65 20 61 6e 73 |ect, but| the ans|
|000005a0| 77 65 72 20 66 6f 72 20 | 11 33 66 20 11 34 6f 20 |wer for |.3f .4o |
|000005b0| 11 33 67 20 11 31 69 73 | 20 6e 6f 74 2e 20 20 57 |.3g .1is| not.  W|
|000005c0| 65 0d 0a 00 66 69 6e 64 | 20 11 33 66 20 11 34 6f |e...find| .3f .4o|
|000005d0| 20 11 33 67 20 11 31 61 | 73 20 66 6f 6c 6c 6f 77 | .3g .1a|s follow|
|000005e0| 73 2e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |s..0... |        |
|000005f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000600| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000610| 20 11 34 67 32 32 32 32 | 32 32 32 32 32 20 20 20 | .4g2222|22222   |
|00000620| 20 20 67 32 32 32 32 32 | 0d 0b 00 20 20 20 20 20 |  g22222|...     |
|00000630| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000640| 20 20 20 20 20 20 20 20 | 20 20 11 32 32 20 20 20 |        |  .22   |
|00000650| 20 20 20 20 20 20 11 34 | 66 20 11 32 32 20 20 20 |      .4|f .22   |
|00000660| 20 20 20 20 20 20 20 20 | 20 11 34 66 20 11 32 32 |        | .4f .22|
|00000670| 0d 0b 00 20 20 20 20 20 | 11 31 28 11 33 66 20 11 |...     |.1(.3f .|
|00000680| 34 6f 20 11 33 67 11 31 | 29 28 11 33 78 11 31 29 |4o .3g.1|)(.3x.1)|
|00000690| 20 3d 20 11 33 66 11 31 | 28 11 33 67 11 31 28 11 | = .3f.1|(.3g.1(.|
|000006a0| 33 78 11 31 29 29 20 3d | 20 11 33 66 11 31 28 11 |3x.1)) =| .3f.1(.|
|000006b0| 33 78 20 20 11 31 2d 20 | 31 29 20 3d 20 11 34 76 |3x  .1- |1) = .4v|
|000006c0| 20 11 33 78 20 20 11 31 | 2d 20 31 20 2b 20 33 20 | .3x  .1|- 1 + 3 |
|000006d0| 3d 20 11 34 76 20 11 33 | 78 20 20 11 31 2b 20 32 |= .4v .3|x  .1+ 2|
|000006e0| 0d 0a 00 00 17 4f 04 0d | 0b 00 57 72 6f 6e 67 2e |.....O..|..Wrong.|
|000006f0| 20 20 59 6f 75 20 6d 75 | 6c 74 69 70 6c 69 65 64 |  You mu|ltiplied|
|00000700| 20 69 6e 73 74 65 61 64 | 20 6f 66 20 66 6f 72 6d | instead| of form|
|00000710| 69 6e 67 20 63 6f 6d 70 | 6f 73 69 74 65 20 66 75 |ing comp|osite fu|
|00000720| 6e 63 74 69 6f 6e 73 2e | 20 20 52 65 76 69 65 77 |nctions.|  Review|
|00000730| 0d 0a 00 47 75 69 64 65 | 64 20 45 78 61 6d 70 6c |...Guide|d Exampl|
|00000740| 65 20 35 20 61 6e 64 20 | 74 72 79 20 61 67 61 69 |e 5 and |try agai|
|00000750| 6e 2e 0d 0a 00 00 17 4f | 03 0d 0b 00 52 69 67 68 |n......O|....Righ|
|00000760| 74 2e 20 20 59 6f 75 20 | 66 6f 75 6e 64 20 11 33 |t.  You |found .3|
|00000770| 66 20 11 34 6f 20 11 33 | 67 20 11 31 61 6e 64 20 |f .4o .3|g .1and |
|00000780| 11 33 67 20 11 34 6f 20 | 11 33 66 20 11 31 63 6f |.3g .4o |.3f .1co|
|00000790| 72 72 65 63 74 6c 79 2e | 0d 0a 00 00 17 4f 03 0d |rrectly.|.....O..|
|000007a0| 0b 00 4f 6f 70 73 21 20 | 20 59 6f 75 20 68 61 76 |..Oops! | You hav|
|000007b0| 65 20 74 68 65 20 63 6f | 6d 70 6f 73 69 74 65 20 |e the co|mposite |
|000007c0| 66 75 6e 63 74 69 6f 6e | 73 20 11 33 66 20 11 34 |function|s .3f .4|
|000007d0| 6f 20 11 33 67 20 11 31 | 61 6e 64 20 11 33 67 20 |o .3g .1|and .3g |
|000007e0| 11 34 6f 20 11 33 66 20 | 11 31 6d 69 78 65 64 20 |.4o .3f |.1mixed |
|000007f0| 75 70 2e 0d 0a 00 00 17 | 4f 07 0d 0b 00 57 72 6f |up......|O....Wro|
|00000800| 6e 67 2e 20 20 57 65 20 | 66 69 6e 64 20 11 33 66 |ng.  We |find .3f|
|00000810| 20 11 34 6f 20 11 33 67 | 20 11 31 61 73 20 66 6f | .4o .3g| .1as fo|
|00000820| 6c 6c 6f 77 73 2e 12 30 | 0d 0a 00 0d 0b 00 20 20 |llows..0|......  |
|00000830| 20 20 20 28 11 33 66 20 | 11 34 6f 20 11 33 67 11 |   (.3f |.4o .3g.|
|00000840| 31 29 28 11 33 78 11 31 | 29 20 3d 20 11 33 66 11 |1)(.3x.1|) = .3f.|
|00000850| 31 28 11 33 67 11 31 28 | 11 33 78 11 31 29 29 20 |1(.3g.1(|.3x.1)) |
|00000860| 3d 20 11 33 66 11 31 28 | 11 33 78 20 11 31 2b 20 |= .3f.1(|.3x .1+ |
|00000870| 32 29 20 3d 20 7c 11 33 | 78 20 11 31 2b 20 32 20 |2) = |.3|x .1+ 2 |
|00000880| 2d 20 34 7c 20 3d 20 7c | 11 33 78 20 11 31 2d 20 |- 4| = ||.3x .1- |
|00000890| 32 7c 0d 0a 00 0d 0b 00 | 57 65 20 66 69 6e 64 20 |2|......|We find |
|000008a0| 11 33 67 20 11 34 6f 20 | 11 33 66 20 11 31 61 73 |.3g .4o |.3f .1as|
|000008b0| 20 66 6f 6c 6c 6f 77 73 | 2e 12 30 0d 0a 00 0d 0b | follows|..0.....|
|000008c0| 00 20 20 20 20 20 28 11 | 33 67 20 11 34 6f 20 11 |.     (.|3g .4o .|
|000008d0| 33 66 11 31 29 28 11 33 | 78 11 31 29 20 3d 20 11 |3f.1)(.3|x.1) = .|
|000008e0| 33 67 11 31 28 11 33 66 | 11 31 28 11 33 78 11 31 |3g.1(.3f|.1(.3x.1|
|000008f0| 29 29 20 3d 20 11 33 67 | 11 31 28 7c 11 33 78 20 |)) = .3g|.1(|.3x |
|00000900| 11 31 2d 20 34 7c 29 20 | 3d 20 7c 11 33 78 20 11 |.1- 4|) |= |.3x .|
|00000910| 31 2d 20 34 7c 20 2b 20 | 32 0d 0a 00 00 17 4f 05 |1- 4| + |2.....O.|
|00000920| 0d 0b 00 57 72 6f 6e 67 | 2e 20 20 54 68 69 73 20 |...Wrong|.  This |
|00000930| 69 73 20 74 68 65 20 64 | 6f 6d 61 69 6e 20 6f 66 |is the d|omain of|
|00000940| 20 11 33 66 11 31 2e 20 | 20 46 69 6e 64 20 74 68 | .3f.1. | Find th|
|00000950| 65 20 64 6f 6d 61 69 6e | 20 6f 66 20 11 33 66 20 |e domain| of .3f |
|00000960| 11 34 6f 20 11 33 67 11 | 31 2c 20 77 68 65 72 65 |.4o .3g.|1, where|
|00000970| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00000980| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 35 0d 0b |        |     5..|
|00000990| 00 20 20 20 20 20 20 20 | 20 20 28 11 33 66 20 11 |.       |  (.3f .|
|000009a0| 34 6f 20 11 33 67 11 31 | 29 28 11 33 78 11 31 29 |4o .3g.1|)(.3x.1)|
|000009b0| 20 3d 20 11 34 32 32 32 | 32 32 32 32 32 11 31 2e | = .4222|22222.1.|
|000009c0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|000009d0| 20 20 20 20 20 20 20 20 | 20 11 33 78 11 31 28 11 |        | .3x.1(.|
|000009e0| 33 78 20 11 31 2b 20 34 | 29 0d 0a 00 00 17 4f 03 |3x .1+ 4|).....O.|
|000009f0| 0d 0b 00 52 69 67 68 74 | 2e 20 20 54 68 65 20 64 |...Right|.  The d|
|00000a00| 6f 6d 61 69 6e 20 6f 66 | 20 11 33 66 20 11 34 6f |omain of| .3f .4o|
|00000a10| 20 11 33 67 20 11 31 69 | 73 20 61 6c 6c 20 72 65 | .3g .1i|s all re|
|00000a20| 61 6c 20 6e 75 6d 62 65 | 72 73 20 65 78 63 65 70 |al numbe|rs excep|
|00000a30| 74 20 66 6f 72 20 11 33 | 78 20 11 31 3d 20 2d 34 |t for .3|x .1= -4|
|00000a40| 20 61 6e 64 20 11 33 78 | 20 11 31 3d 20 30 2e 0d | and .3x| .1= 0..|
|00000a50| 0a 00 00 17 4f 08 0d 0b | 00 57 72 6f 6e 67 2e 20 |....O...|.Wrong. |
|00000a60| 20 53 69 6e 63 65 20 11 | 33 66 20 11 34 6f 20 11 | Since .|3f .4o .|
|00000a70| 33 67 20 11 31 69 73 20 | 67 69 76 65 6e 20 62 79 |3g .1is |given by|
|00000a80| 20 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 | ...    |        |
|00000a90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 35 0d |        |      5.|
|00000aa0| 0b 00 20 20 20 20 20 20 | 20 20 20 28 11 33 66 20 |..      |   (.3f |
|00000ab0| 11 34 6f 20 11 33 67 11 | 31 29 28 11 33 78 11 31 |.4o .3g.|1)(.3x.1|
|00000ac0| 29 20 3d 20 11 34 32 32 | 32 32 32 32 32 32 20 0d |) = .422|222222 .|
|00000ad0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00000ae0| 20 20 20 20 20 20 20 20 | 11 33 78 11 31 28 11 33 |        |.3x.1(.3|
|00000af0| 78 20 11 31 2b 20 34 29 | 0d 0a 00 77 65 20 63 6f |x .1+ 4)|...we co|
|00000b00| 6e 63 6c 75 64 65 20 74 | 68 61 74 20 74 68 65 20 |nclude t|hat the |
|00000b10| 64 6f 6d 61 69 6e 20 6f | 66 20 11 33 66 20 11 34 |domain o|f .3f .4|
|00000b20| 6f 20 11 33 67 20 11 31 | 69 73 20 61 6c 6c 20 72 |o .3g .1|is all r|
|00000b30| 65 61 6c 20 6e 75 6d 62 | 65 72 73 20 65 78 63 65 |eal numb|ers exce|
|00000b40| 70 74 20 66 6f 72 20 11 | 33 78 20 11 31 3d 20 2d |pt for .|3x .1= -|
|00000b50| 34 0d 0a 00 61 6e 64 20 | 11 33 78 20 11 31 3d 20 |4...and |.3x .1= |
|00000b60| 30 2e 0d 0a 00 00 17 4f | 08 0d 0b 00 57 72 6f 6e |0......O|....Wron|
|00000b70| 67 2e 20 20 53 69 6e 63 | 65 20 11 33 66 20 11 34 |g.  Sinc|e .3f .4|
|00000b80| 6f 20 11 33 67 20 11 31 | 69 73 20 67 69 76 65 6e |o .3g .1|is given|
|00000b90| 20 62 79 20 0d 0a 00 20 | 20 20 20 20 20 20 20 20 | by ... |        |
|00000ba0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000bb0| 20 35 0d 0b 00 20 20 20 | 20 20 20 20 20 20 28 11 | 5...   |      (.|
|00000bc0| 33 66 20 11 34 6f 20 11 | 33 67 11 31 29 28 11 33 |3f .4o .|3g.1)(.3|
|00000bd0| 78 11 31 29 20 3d 20 11 | 34 32 32 32 32 32 32 32 |x.1) = .|42222222|
|00000be0| 32 20 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |2 ...   |        |
|00000bf0| 20 20 20 20 20 20 20 20 | 20 20 20 11 33 78 11 31 |        |   .3x.1|
|00000c00| 28 11 33 78 20 11 31 2b | 20 34 29 0d 0a 00 77 65 |(.3x .1+| 4)...we|
|00000c10| 20 63 6f 6e 63 6c 75 64 | 65 20 74 68 61 74 20 74 | conclud|e that t|
|00000c20| 68 65 20 64 6f 6d 61 69 | 6e 20 6f 66 20 11 33 66 |he domai|n of .3f|
|00000c30| 20 11 34 6f 20 11 33 67 | 20 11 31 69 73 20 61 6c | .4o .3g| .1is al|
|00000c40| 6c 20 72 65 61 6c 20 6e | 75 6d 62 65 72 73 20 65 |l real n|umbers e|
|00000c50| 78 63 65 70 74 20 66 6f | 72 20 11 33 78 20 11 31 |xcept fo|r .3x .1|
|00000c60| 3d 20 2d 34 0d 0a 00 61 | 6e 64 20 11 33 78 20 11 |= -4...a|nd .3x .|
|00000c70| 31 3d 20 30 2e 0d 0a 00 | 00 17 4f 06 20 20 20 20 |1= 0....|..O.    |
|00000c80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000c90| 20 20 20 20 20 11 32 32 | 0d 0b 00 11 31 53 6f 72 |     .22|....1Sor|
|00000ca0| 72 79 2c 20 62 75 74 20 | 28 11 33 66 20 11 31 2b |ry, but |(.3f .1+|
|00000cb0| 20 11 33 67 11 31 29 28 | 11 33 78 11 31 29 20 11 | .3g.1)(|.3x.1) .|
|00000cc0| 34 3d 20 11 33 78 20 20 | 11 31 2b 20 34 11 33 78 |4= .3x  |.1+ 4.3x|
|00000cd0| 20 11 31 2b 20 36 2e 20 | 20 57 65 20 66 69 6e 64 | .1+ 6. | We find|
|00000ce0| 20 28 11 33 66 20 11 31 | 2b 20 11 33 67 11 31 29 | (.3f .1|+ .3g.1)|
|00000cf0| 28 11 33 78 11 31 29 20 | 61 73 20 66 6f 6c 6c 6f |(.3x.1) |as follo|
|00000d00| 77 73 2e 12 30 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |ws..0...|...     |
|00000d10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000d20| 20 20 20 20 20 20 20 20 | 20 20 11 32 32 20 20 20 |        |  .22   |
|00000d30| 20 20 20 20 20 20 32 0d | 0b 00 20 20 20 20 20 11 |      2.|..     .|
|00000d40| 31 28 11 33 66 20 11 31 | 2b 20 11 33 67 11 31 29 |1(.3f .1|+ .3g.1)|
|00000d50| 28 11 33 78 11 31 29 20 | 3d 20 28 34 11 33 78 20 |(.3x.1) |= (4.3x |
|00000d60| 11 31 2d 20 33 29 20 2b | 20 28 11 33 78 20 20 11 |.1- 3) +| (.3x  .|
|00000d70| 31 2d 20 39 29 20 3d 20 | 11 33 78 20 20 11 31 2b |1- 9) = |.3x  .1+|
|00000d80| 20 34 11 33 78 20 11 31 | 2d 20 31 32 20 20 0d 0a | 4.3x .1|- 12  ..|
|00000d90| 00 0d 0b 00 54 72 79 20 | 61 67 61 69 6e 2e 20 20 |....Try |again.  |
|00000da0| 20 12 30 0d 0a 00 00 17 | 4f 06 20 20 20 20 20 20 | .0.....|O.      |
|00000db0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000dc0| 20 20 20 20 11 32 32 0d | 0b 00 11 31 53 6f 72 72 |    .22.|...1Sorr|
|00000dd0| 79 2c 20 62 75 74 20 28 | 11 33 66 20 11 31 2d 20 |y, but (|.3f .1- |
|00000de0| 11 33 67 11 31 29 28 11 | 33 78 11 31 29 20 11 34 |.3g.1)(.|3x.1) .4|
|00000df0| 3d 20 11 31 2d 11 33 78 | 20 20 11 31 2b 20 34 11 |= .1-.3x|  .1+ 4.|
|00000e00| 33 78 20 11 31 2d 20 31 | 32 2e 20 20 57 65 20 66 |3x .1- 1|2.  We f|
|00000e10| 69 6e 64 20 28 11 33 66 | 20 11 31 2d 20 11 33 67 |ind (.3f| .1- .3g|
|00000e20| 11 31 29 28 11 33 78 11 | 31 29 20 61 73 20 66 6f |.1)(.3x.|1) as fo|
|00000e30| 6c 6c 6f 77 73 2e 12 30 | 0d 0a 00 0d 0b 00 20 20 |llows..0|......  |
|00000e40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e50| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 32 |        |     .22|
|00000e60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000e70| 20 20 32 20 20 20 20 20 | 20 20 20 20 32 0d 0b 00 |  2     |    2...|
|00000e80| 20 20 20 20 20 11 31 28 | 11 33 66 20 11 31 2d 20 |     .1(|.3f .1- |
|00000e90| 11 33 67 11 31 29 28 11 | 33 78 11 31 29 20 3d 20 |.3g.1)(.|3x.1) = |
|00000ea0| 28 34 11 33 78 20 11 31 | 2d 20 33 29 20 2d 20 28 |(4.3x .1|- 3) - (|
|00000eb0| 11 33 78 20 20 11 31 2d | 20 39 29 20 3d 20 34 11 |.3x  .1-| 9) = 4.|
|00000ec0| 33 78 20 11 31 2d 20 33 | 20 2d 20 11 33 78 20 20 |3x .1- 3| - .3x  |
|00000ed0| 11 31 2b 20 39 20 3d 20 | 2d 11 33 78 20 20 11 31 |.1+ 9 = |-.3x  .1|
|00000ee0| 2b 20 34 11 33 78 20 11 | 31 2b 20 36 20 20 0d 0a |+ 4.3x .|1+ 6  ..|
|00000ef0| 00 0d 0b 00 54 72 79 20 | 61 67 61 69 6e 2e 12 30 |....Try |again..0|
|00000f00| 0d 0a 00 00 17 4f 09 20 | 20 20 20 20 20 20 20 20 |.....O. |        |
|00000f10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 34 |        |       4|
|00000f20| 0d 0b 00 53 6f 72 72 79 | 2c 20 62 75 74 20 28 11 |...Sorry|, but (.|
|00000f30| 33 66 11 31 2f 11 33 67 | 11 31 29 28 11 33 78 11 |3f.1/.3g|.1)(.3x.|
|00000f40| 31 29 20 11 34 3d 20 32 | 32 32 32 32 11 31 2e 20 |1) .4= 2|2222.1. |
|00000f50| 20 57 65 20 66 69 6e 64 | 20 28 11 33 66 11 31 2f | We find| (.3f.1/|
|00000f60| 11 33 67 11 31 29 28 11 | 33 78 11 31 29 20 61 73 |.3g.1)(.|3x.1) as|
|00000f70| 20 66 6f 6c 6c 6f 77 73 | 2e 0d 0b 00 20 20 20 20 | follows|....    |
|00000f80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000f90| 20 20 11 33 78 20 11 31 | 2d 20 33 20 20 20 20 20 |  .3x .1|- 3     |
|00000fa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000fb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 12 30 |        |      .0|
|00000fc0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|00000fd0| 20 20 20 20 20 20 34 11 | 33 78 20 11 31 2d 20 33 |      4.|3x .1- 3|
|00000fe0| 0d 0b 00 20 20 20 20 20 | 28 11 33 66 11 31 2f 11 |...     |(.3f.1/.|
|00000ff0| 33 67 11 31 29 28 11 33 | 78 11 31 29 20 3d 20 11 |3g.1)(.3|x.1) = .|
|00001000| 34 32 32 32 32 32 32 11 | 31 2c 20 11 33 78 20 11 |4222222.|1, .3x .|
|00001010| 34 3d 20 2b 11 31 33 0d | 0b 00 20 20 20 20 20 20 |4= +.13.|..      |
|00001020| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|00001030| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001040| 20 11 33 78 20 20 11 31 | 2d 20 39 20 20 20 20 20 | .3x  .1|- 9     |
|00001050| 20 20 20 20 0d 0a 00 0d | 0b 00 54 72 79 20 61 67 |    ....|..Try ag|
|00001060| 61 69 6e 2e 12 30 0d 0a | 00 00 17 4f 04 20 20 20 |ain..0..|...O.   |
|00001070| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001080| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001090| 20 20 11 32 33 20 20 20 | 20 20 32 0d 0b 00 11 31 |  .23   |  2....1|
|000010a0| 52 69 67 68 74 2e 20 20 | 54 68 65 20 63 6f 6d 62 |Right.  |The comb|
|000010b0| 69 6e 61 74 69 6f 6e 20 | 28 11 33 66 67 11 31 29 |ination |(.3fg.1)|
|000010c0| 28 11 33 78 11 31 29 20 | 69 73 20 34 11 33 78 20 |(.3x.1) |is 4.3x |
|000010d0| 20 11 31 2d 20 33 11 33 | 78 20 20 11 31 2d 20 33 | .1- 3.3|x  .1- 3|
|000010e0| 36 11 33 78 20 11 31 2b | 20 32 37 2e 20 20 54 68 |6.3x .1+| 27.  Th|
|000010f0| 65 20 64 6f 6d 61 69 6e | 20 69 73 20 61 6c 6c 0d |e domain| is all.|
|00001100| 0a 00 72 65 61 6c 20 6e | 75 6d 62 65 72 73 2e 0d |..real n|umbers..|
|00001110| 0a 00 00 17 4f 08 0d 0b | 00 53 6f 72 72 79 2c 20 |....O...|.Sorry, |
|00001120| 62 75 74 20 28 11 33 66 | 20 11 31 2b 20 11 33 67 |but (.3f| .1+ .3g|
|00001130| 11 31 29 28 34 29 20 11 | 34 3d 20 11 31 32 31 2e |.1)(4) .|4= .121.|
|00001140| 20 20 46 69 72 73 74 2c | 20 77 65 20 66 69 6e 64 |  First,| we find|
|00001150| 20 28 11 33 66 20 11 31 | 2b 20 11 33 67 11 31 29 | (.3f .1|+ .3g.1)|
|00001160| 28 11 33 78 11 31 29 20 | 61 73 20 66 6f 6c 6c 6f |(.3x.1) |as follo|
|00001170| 77 73 2e 12 30 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |ws..0...|...     |
|00001180| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001190| 34 44 32 20 20 20 20 20 | 20 11 32 32 20 20 20 20 |4D2     | .22    |
|000011a0| 20 20 20 20 20 32 20 20 | 20 20 11 34 44 32 0d 0b |     2  |  .4D2..|
|000011b0| 00 20 20 20 20 20 11 31 | 28 11 33 66 20 11 31 2b |.     .1|(.3f .1+|
|000011c0| 20 11 33 67 11 31 29 28 | 11 33 78 11 31 29 20 3d | .3g.1)(|.3x.1) =|
|000011d0| 20 28 11 34 53 20 11 33 | 78 11 31 29 20 2b 20 28 | (.4S .3|x.1) + (|
|000011e0| 11 33 78 20 20 11 31 2b | 20 31 29 20 3d 20 11 33 |.3x  .1+| 1) = .3|
|000011f0| 78 20 20 11 31 2b 20 11 | 34 53 20 11 33 78 20 11 |x  .1+ .|4S .3x .|
|00001200| 31 2b 20 31 12 30 0d 0a | 00 0d 0b 00 57 65 20 66 |1+ 1.0..|....We f|
|00001210| 69 6e 64 20 28 11 33 66 | 20 11 31 2b 20 11 33 67 |ind (.3f| .1+ .3g|
|00001220| 11 31 29 28 34 29 20 61 | 73 20 66 6f 6c 6c 6f 77 |.1)(4) a|s follow|
|00001230| 73 2e 12 30 0d 0a 00 0d | 0b 00 20 20 20 20 20 20 |s..0....|..      |
|00001240| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001250| 32 32 20 20 20 20 11 34 | 44 32 0d 0b 00 20 20 20 |22    .4|D2...   |
|00001260| 20 20 11 31 28 11 33 66 | 20 11 31 2b 20 11 33 67 |  .1(.3f| .1+ .3g|
|00001270| 11 31 29 28 34 29 20 3d | 20 28 34 29 20 20 2b 20 |.1)(4) =| (4)  + |
|00001280| 11 34 53 20 11 31 34 20 | 2b 20 31 20 3d 20 31 36 |.4S .14 |+ 1 = 16|
|00001290| 20 2b 20 32 20 2b 20 31 | 20 3d 20 31 39 12 30 0d | + 2 + 1| = 19.0.|
|000012a0| 0a 00 00 17 4f 08 0d 0b | 00 53 6f 72 72 79 2c 20 |....O...|.Sorry, |
|000012b0| 62 75 74 20 28 11 33 66 | 20 11 31 2d 20 11 33 67 |but (.3f| .1- .3g|
|000012c0| 11 31 29 28 39 29 20 11 | 34 3d 20 11 31 2d 37 37 |.1)(9) .|4= .1-77|
|000012d0| 2e 20 20 46 69 72 73 74 | 2c 20 77 65 20 66 69 6e |.  First|, we fin|
|000012e0| 64 20 28 11 33 66 20 11 | 31 2d 20 11 33 67 11 31 |d (.3f .|1- .3g.1|
|000012f0| 29 28 11 33 78 11 31 29 | 20 61 73 20 66 6f 6c 6c |)(.3x.1)| as foll|
|00001300| 6f 77 73 2e 12 30 0d 0a | 00 0d 0b 00 20 20 20 20 |ows..0..|....    |
|00001310| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001320| 11 34 44 32 20 20 20 20 | 20 20 11 32 32 20 20 20 |.4D2    |  .22   |
|00001330| 20 20 20 20 20 20 11 34 | 44 32 20 20 20 20 11 32 |      .4|D2    .2|
|00001340| 32 0d 0b 00 20 20 20 20 | 20 11 31 28 11 33 66 20 |2...    | .1(.3f |
|00001350| 11 31 2d 20 11 33 67 11 | 31 29 28 11 33 78 11 31 |.1- .3g.|1)(.3x.1|
|00001360| 29 20 3d 20 28 11 34 53 | 20 11 33 78 11 31 29 20 |) = (.4S| .3x.1) |
|00001370| 2d 20 28 11 33 78 20 20 | 11 31 2b 20 31 29 20 3d |- (.3x  |.1+ 1) =|
|00001380| 20 11 34 53 20 11 33 78 | 20 11 31 2d 20 11 33 78 | .4S .3x| .1- .3x|
|00001390| 20 20 11 31 2d 20 31 12 | 30 0d 0a 00 0d 0b 00 57 |  .1- 1.|0......W|
|000013a0| 65 20 66 69 6e 64 20 28 | 11 33 66 20 11 31 2d 20 |e find (|.3f .1- |
|000013b0| 11 33 67 11 31 29 28 39 | 29 20 61 73 20 66 6f 6c |.3g.1)(9|) as fol|
|000013c0| 6c 6f 77 73 2e 12 30 0d | 0a 00 0d 0b 00 20 20 20 |lows..0.|.....   |
|000013d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000013e0| 11 34 44 32 20 20 20 20 | 20 20 11 32 32 0d 0b 00 |.4D2    |  .22...|
|000013f0| 20 20 20 20 20 11 31 28 | 11 33 66 20 11 31 2d 20 |     .1(|.3f .1- |
|00001400| 11 33 67 11 31 29 28 39 | 29 20 3d 20 11 34 53 20 |.3g.1)(9|) = .4S |
|00001410| 11 31 39 20 2d 20 28 39 | 29 20 20 2d 20 31 20 3d |.19 - (9|)  - 1 =|
|00001420| 20 33 20 2d 20 38 31 20 | 2d 20 31 20 3d 20 2d 37 | 3 - 81 |- 1 = -7|
|00001430| 39 20 20 12 30 0d 0a 00 | 00 17 4f 07 0d 0b 00 53 |9  .0...|..O....S|
|00001440| 6f 72 72 79 2c 20 62 75 | 74 20 28 11 33 66 20 11 |orry, bu|t (.3f .|
|00001450| 31 2b 20 11 33 67 11 31 | 29 28 2d 31 29 20 11 34 |1+ .3g.1|)(-1) .4|
|00001460| 3d 20 11 31 32 31 2e 20 | 20 46 69 72 73 74 2c 20 |= .121. | First, |
|00001470| 77 65 20 66 69 6e 64 20 | 28 11 33 66 20 11 31 2b |we find |(.3f .1+|
|00001480| 20 11 33 67 11 31 29 28 | 11 33 78 11 31 29 20 61 | .3g.1)(|.3x.1) a|
|00001490| 73 20 66 6f 6c 6c 6f 77 | 73 2e 12 30 0d 0a 00 0d |s follow|s..0....|
|000014a0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000014b0| 20 20 20 20 20 20 11 34 | 44 32 20 20 20 20 20 20 |      .4|D2      |
|000014c0| 11 32 32 20 20 20 20 20 | 20 20 20 20 32 20 20 20 |.22     |    2   |
|000014d0| 20 11 34 44 32 0d 0b 00 | 20 20 20 20 20 11 31 28 | .4D2...|     .1(|
|000014e0| 11 33 66 20 11 31 2b 20 | 11 33 67 11 31 29 28 11 |.3f .1+ |.3g.1)(.|
|000014f0| 33 78 11 31 29 20 3d 20 | 28 11 34 53 20 11 33 78 |3x.1) = |(.4S .3x|
|00001500| 11 31 29 20 2b 20 28 11 | 33 78 20 20 11 31 2b 20 |.1) + (.|3x  .1+ |
|00001510| 31 29 20 3d 20 11 33 78 | 20 20 11 31 2b 20 11 34 |1) = .3x|  .1+ .4|
|00001520| 53 20 11 33 78 20 11 31 | 2b 20 31 0d 0a 00 20 20 |S .3x .1|+ 1...  |
|00001530| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 34 |        |      .4|
|00001540| 44 32 0d 0b 00 11 31 53 | 69 6e 63 65 20 74 68 65 |D2....1S|ince the|
|00001550| 20 74 65 72 6d 20 11 34 | 53 20 11 33 78 20 11 31 | term .4|S .3x .1|
|00001560| 69 6d 70 6c 69 65 73 20 | 74 68 61 74 20 11 33 78 |implies |that .3x|
|00001570| 20 11 34 3e 20 11 31 30 | 2c 20 77 65 20 63 6f 6e | .4> .10|, we con|
|00001580| 63 6c 75 64 65 20 74 68 | 61 74 20 69 74 20 69 73 |clude th|at it is|
|00001590| 20 6e 6f 74 20 0d 0a 00 | 70 6f 73 73 69 62 6c 65 | not ...|possible|
|000015a0| 20 74 6f 20 66 69 6e 64 | 20 28 11 33 66 20 11 31 | to find| (.3f .1|
|000015b0| 2b 20 11 33 67 11 31 29 | 28 2d 31 29 2e 12 30 0d |+ .3g.1)|(-1)..0.|
|000015c0| 0a 00 00 17 4f 03 20 20 | 20 20 20 20 20 20 20 20 |....O.  |        |
|000015d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000015e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 34 44 32 |        |    .4D2|
|000015f0| 20 20 20 20 20 20 11 32 | 32 0d 0b 00 11 31 52 69 |      .2|2....1Ri|
|00001600| 67 68 74 2e 20 20 54 68 | 65 20 63 6f 6d 70 6f 73 |ght.  Th|e compos|
|00001610| 69 74 69 6f 6e 20 28 11 | 33 66 20 11 31 2d 20 11 |ition (.|3f .1- .|
|00001620| 33 67 11 31 29 28 31 29 | 20 3d 20 11 34 53 20 11 |3g.1)(1)| = .4S .|
|00001630| 31 31 20 2d 20 28 31 29 | 20 20 2d 20 31 20 3d 20 |11 - (1)|  - 1 = |
|00001640| 31 20 2d 20 31 20 2d 20 | 31 20 3d 20 2d 31 2e 0d |1 - 1 - |1 = -1..|
|00001650| 0a 00 00 14 4f 04 0d 0b | 00 59 65 73 20 61 6e 64 |....O...|.Yes and|
|00001660| 20 6e 6f 2e 20 20 54 68 | 69 73 20 69 73 20 63 6f | no.  Th|is is co|
|00001670| 72 72 65 63 74 2c 20 62 | 75 74 20 73 6f 20 61 72 |rrect, b|ut so ar|
|00001680| 65 20 28 62 29 20 61 6e | 64 20 28 63 29 2c 20 73 |e (b) an|d (c), s|
|00001690| 6f 20 74 68 65 20 62 65 | 73 74 20 61 6e 73 77 65 |o the be|st answe|
|000016a0| 72 20 69 73 0d 0a 00 28 | 64 29 2e 0d 0a 00 00 14 |r is...(|d)......|
|000016b0| 4f 04 0d 0b 00 59 65 73 | 20 61 6e 64 20 6e 6f 2e |O....Yes| and no.|
|000016c0| 20 20 54 68 69 73 20 69 | 73 20 63 6f 72 72 65 63 |  This i|s correc|
|000016d0| 74 2c 20 62 75 74 20 73 | 6f 20 61 72 65 20 28 61 |t, but s|o are (a|
|000016e0| 29 20 61 6e 64 20 28 63 | 29 2c 20 73 6f 20 74 68 |) and (c|), so th|
|000016f0| 65 20 62 65 73 74 20 61 | 6e 73 77 65 72 20 69 73 |e best a|nswer is|
|00001700| 0d 0a 00 28 64 29 2e 0d | 0a 00 00 14 4f 04 0d 0b |...(d)..|....O...|
|00001710| 00 59 65 73 20 61 6e 64 | 20 6e 6f 2e 20 20 54 68 |.Yes and| no.  Th|
|00001720| 69 73 20 69 73 20 63 6f | 72 72 65 63 74 2c 20 62 |is is co|rrect, b|
|00001730| 75 74 20 73 6f 20 61 72 | 65 20 28 61 29 20 61 6e |ut so ar|e (a) an|
|00001740| 64 20 28 62 29 2c 20 73 | 6f 20 74 68 65 20 62 65 |d (b), s|o the be|
|00001750| 73 74 20 61 6e 73 77 65 | 72 20 69 73 0d 0a 00 28 |st answe|r is...(|
|00001760| 64 29 2e 0d 0a 00 00 14 | 4f 03 0d 0b 00 47 6f 6f |d)......|O....Goo|
|00001770| 64 21 20 20 54 68 65 72 | 65 20 61 72 65 20 6d 61 |d!  Ther|e are ma|
|00001780| 6e 79 20 77 61 79 73 20 | 74 68 61 74 20 61 20 66 |ny ways |that a f|
|00001790| 75 6e 63 74 69 6f 6e 20 | 63 61 6e 20 62 65 20 22 |unction |can be "|
|000017a0| 64 65 63 6f 6d 70 6f 73 | 65 64 22 2e 0d 0a 00 00 |decompos|ed".....|
|000017b0| 14 4f 05 0d 0b 00 43 6f | 72 72 65 63 74 21 20 20 |.O....Co|rrect!  |
|000017c0| 57 65 20 63 61 6e 20 76 | 65 72 69 66 79 20 74 68 |We can v|erify th|
|000017d0| 69 73 20 22 64 65 63 6f | 6d 70 6f 73 69 74 69 6f |is "deco|mpositio|
|000017e0| 6e 22 20 61 73 20 66 6f | 6c 6c 6f 77 73 2e 12 30 |n" as fo|llows..0|
|000017f0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|00001800| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001810| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001820| 20 31 0d 0b 00 20 20 20 | 20 20 28 11 33 66 20 11 | 1...   |  (.3f .|
|00001830| 34 6f 20 11 33 67 11 31 | 29 28 11 33 78 11 31 29 |4o .3g.1|)(.3x.1)|
|00001840| 20 3d 20 11 33 66 11 31 | 28 11 33 67 11 31 28 11 | = .3f.1|(.3g.1(.|
|00001850| 33 78 11 31 29 29 20 3d | 20 11 33 66 11 31 28 33 |3x.1)) =| .3f.1(3|
|00001860| 11 33 78 20 11 31 2d 20 | 31 29 20 3d 20 11 34 32 |.3x .1- |1) = .42|
|00001870| 32 32 32 32 32 11 31 2e | 0d 0b 00 20 20 20 20 20 |22222.1.|...     |
|00001880| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001890| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000018a0| 20 20 20 33 11 33 78 20 | 11 31 2d 20 31 0d 0a 00 |   3.3x |.1- 1...|
|000018b0| 00 14 4f 06 0d 0b 00 4e | 6f 2e 20 20 54 68 65 20 |..O....N|o.  The |
|000018c0| 63 6f 6d 70 6f 73 69 74 | 69 6f 6e 20 6f 66 20 74 |composit|ion of t|
|000018d0| 68 65 73 65 20 74 77 6f | 20 66 75 6e 63 74 69 6f |hese two| functio|
|000018e0| 6e 73 20 77 6f 75 6c 64 | 20 62 65 0d 0a 00 0d 0b |ns would| be.....|
|000018f0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001910| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001920| 20 20 20 33 20 2d 20 11 | 33 78 0d 0b 00 20 20 20 |   3 - .|3x...   |
|00001930| 20 20 11 31 28 11 33 66 | 20 11 34 6f 20 11 33 67 |  .1(.3f| .4o .3g|
|00001940| 11 31 29 28 11 33 78 11 | 31 29 20 3d 20 11 33 66 |.1)(.3x.|1) = .3f|
|00001950| 11 31 28 11 33 67 11 31 | 28 11 33 78 11 31 29 29 |.1(.3g.1|(.3x.1))|
|00001960| 20 3d 20 11 33 66 11 31 | 28 31 2f 11 33 78 11 31 | = .3f.1|(1/.3x.1|
|00001970| 29 20 3d 20 33 28 31 2f | 11 33 78 11 31 29 20 2d |) = 3(1/|.3x.1) -|
|00001980| 20 31 20 3d 20 11 34 32 | 32 32 32 32 11 31 2e 0d | 1 = .42|2222.1..|
|00001990| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000019a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000019b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000019c0| 20 20 20 20 20 20 11 33 | 78 0d 0a 00 11 31 54 72 |      .3|x....1Tr|
|000019d0| 79 20 61 67 61 69 6e 2e | 0d 0a 00 00 14 4f 06 0d |y again.|.....O..|
|000019e0| 0b 00 4e 6f 2e 20 20 54 | 68 65 20 63 6f 6d 70 6f |..No.  T|he compo|
|000019f0| 73 69 74 69 6f 6e 20 6f | 66 20 74 68 65 73 65 20 |sition o|f these |
|00001a00| 74 77 6f 20 66 75 6e 63 | 74 69 6f 6e 73 20 77 6f |two func|tions wo|
|00001a10| 75 6c 64 20 62 65 0d 0a | 00 0d 0b 00 20 20 20 20 |uld be..|....    |
|00001a20| 20 28 11 33 66 20 11 34 | 6f 20 11 33 67 11 31 29 | (.3f .4|o .3g.1)|
|00001a30| 28 11 33 78 11 31 29 20 | 3d 20 11 33 66 11 31 28 |(.3x.1) |= .3f.1(|
|00001a40| 11 33 67 11 31 28 11 33 | 78 11 31 29 29 20 3d 20 |.3g.1(.3|x.1)) = |
|00001a50| 11 33 66 11 31 28 31 2f | 28 33 11 33 78 20 11 31 |.3f.1(1/|(3.3x .1|
|00001a60| 2d 20 31 29 29 20 3d 20 | 33 11 33 78 20 11 31 2d |- 1)) = |3.3x .1-|
|00001a70| 20 31 2e 0d 0a 00 54 72 | 79 20 61 67 61 69 6e 2e | 1....Tr|y again.|
|00001a80| 0d 0a 00 00 14 4f 03 0d | 0b 00 4e 6f 2e 20 20 4f |.....O..|..No.  O|
|00001a90| 6e 6c 79 20 6f 6e 65 20 | 6f 66 20 74 68 65 73 65 |nly one |of these|
|00001aa0| 20 69 73 20 63 6f 72 72 | 65 63 74 2e 20 20 54 72 | is corr|ect.  Tr|
|00001ab0| 79 20 61 67 61 69 6e 2e | 0d 0a 00 53 65 63 74 69 |y again.|...Secti|
|00001ac0| 6f 6e 20 32 2e 34 20 20 | 54 72 61 6e 73 6c 61 74 |on 2.4  |Translat|
|00001ad0| 69 6f 6e 73 20 61 6e 64 | 20 43 6f 6d 62 69 6e 61 |ions and| Combina|
|00001ae0| 74 69 6f 6e 73 0d 0a 00 | 0d 0a 00 0d 0b 00 20 20 |tions...|......  |
|00001af0| 20 20 0e 78 32 2d 34 2d | 31 0e 54 75 74 6f 72 69 |  .x2-4-|1.Tutori|
|00001b00| 61 6c 20 45 78 65 72 63 | 69 73 65 20 20 31 0f 0d |al Exerc|ise  1..|
|00001b10| 0a 00 0d 0b 00 20 20 20 | 20 0e 78 32 2d 34 2d 32 |.....   | .x2-4-2|
|00001b20| 0e 54 75 74 6f 72 69 61 | 6c 20 45 78 65 72 63 69 |.Tutoria|l Exerci|
|00001b30| 73 65 20 20 32 0f 20 20 | 20 20 0e 78 32 2d 34 2d |se  2.  |  .x2-4-|
|00001b40| 33 0e 54 75 74 6f 72 69 | 61 6c 20 45 78 65 72 63 |3.Tutori|al Exerc|
|00001b50| 69 73 65 20 20 33 0f 0d | 0a 00 0d 0b 00 20 20 20 |ise  3..|.....   |
|00001b60| 20 0e 78 32 2d 34 2d 34 | 0e 54 75 74 6f 72 69 61 | .x2-4-4|.Tutoria|
|00001b70| 6c 20 45 78 65 72 63 69 | 73 65 20 20 34 0f 0d 0a |l Exerci|se  4...|
|00001b80| 00 0d 0b 00 20 20 20 20 | 0e 78 32 2d 34 2d 35 0e |....    |.x2-4-5.|
|00001b90| 54 75 74 6f 72 69 61 6c | 20 45 78 65 72 63 69 73 |Tutorial| Exercis|
|00001ba0| 65 20 20 35 0f 20 20 20 | 20 0e 78 32 2d 34 2d 36 |e  5.   | .x2-4-6|
|00001bb0| 0e 54 75 74 6f 72 69 61 | 6c 20 45 78 65 72 63 69 |.Tutoria|l Exerci|
|00001bc0| 73 65 20 20 36 0f 0d 0a | 00 0d 0b 00 20 20 20 20 |se  6...|....    |
|00001bd0| 0e 78 32 2d 34 2d 37 0e | 54 75 74 6f 72 69 61 6c |.x2-4-7.|Tutorial|
|00001be0| 20 45 78 65 72 63 69 73 | 65 20 20 37 0f 20 20 20 | Exercis|e  7.   |
|00001bf0| 20 0e 78 32 2d 34 2d 38 | 0e 54 75 74 6f 72 69 61 | .x2-4-8|.Tutoria|
|00001c00| 6c 20 45 78 65 72 63 69 | 73 65 20 20 38 0f 0d 0a |l Exerci|se  8...|
|00001c10| 00 0d 0b 00 20 20 20 20 | 0e 78 32 2d 34 2d 39 0e |....    |.x2-4-9.|
|00001c20| 54 75 74 6f 72 69 61 6c | 20 45 78 65 72 63 69 73 |Tutorial| Exercis|
|00001c30| 65 20 20 39 0f 0d 0a 00 | 0d 0b 00 53 65 63 74 69 |e  9....|...Secti|
|00001c40| 6f 6e 20 32 2e 34 20 20 | 54 72 61 6e 73 6c 61 74 |on 2.4  |Translat|
|00001c50| 69 6f 6e 73 20 61 6e 64 | 20 43 6f 6d 62 69 6e 61 |ions and| Combina|
|00001c60| 74 69 6f 6e 73 20 20 20 | 20 20 20 20 20 20 20 20 |tions   |        |
|00001c70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001c80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001c90| 32 32 0d 0b 00 20 11 31 | 31 2e 20 20 47 69 76 65 |22... .1|1.  Give|
|00001ca0| 6e 20 74 68 61 74 20 11 | 33 66 11 31 28 11 33 78 |n that .|3f.1(.3x|
|00001cb0| 11 31 29 20 3d 20 34 11 | 33 78 20 11 31 2d 20 33 |.1) = 4.|3x .1- 3|
|00001cc0| 20 61 6e 64 20 11 33 67 | 11 31 28 11 33 78 11 31 | and .3g|.1(.3x.1|
|00001cd0| 29 20 3d 20 11 33 78 20 | 20 11 31 2d 20 39 2c 20 |) = .3x | .1- 9, |
|00001ce0| 77 68 69 63 68 20 6f 66 | 20 74 68 65 20 66 6f 6c |which of| the fol|
|00001cf0| 6c 6f 77 69 6e 67 20 69 | 73 20 0d 0a 00 20 20 20 |lowing i|s ...   |
|00001d00| 20 20 11 33 63 6f 72 72 | 65 63 74 11 31 3f 0d 0a |  .3corr|ect.1?..|
|00001d10| 00 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|00001d20| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 20 20 |        |   .22  |
|00001d30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001d40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 32 |        |       2|
|00001d50| 0d 0b 00 20 20 20 20 20 | 11 31 0e 33 2d 36 2d 35 |...     |.1.3-6-5|
|00001d60| 61 0e 28 61 29 0f 20 28 | 11 33 66 20 11 31 2b 20 |a.(a). (|.3f .1+ |
|00001d70| 11 33 67 11 31 29 28 11 | 33 78 11 31 29 20 3d 20 |.3g.1)(.|3x.1) = |
|00001d80| 11 33 78 20 20 11 31 2b | 20 34 11 33 78 20 11 31 |.3x  .1+| 4.3x .1|
|00001d90| 2b 20 36 20 20 20 20 20 | 0e 33 2d 36 2d 35 62 0e |+ 6     |.3-6-5b.|
|00001da0| 28 62 29 0f 20 28 11 33 | 66 20 11 31 2d 20 11 33 |(b). (.3|f .1- .3|
|00001db0| 67 11 31 29 28 11 33 78 | 11 31 29 20 3d 20 2d 11 |g.1)(.3x|.1) = -.|
|00001dc0| 33 78 20 20 11 31 2b 20 | 34 11 33 78 20 11 31 2d |3x  .1+ |4.3x .1-|
|00001dd0| 20 31 32 0d 0a 00 20 20 | 20 20 20 20 20 20 20 44 | 12...  |       D|
|00001de0| 6f 6d 61 69 6e 3a 20 28 | 2d 11 34 38 11 31 2c 20 |omain: (|-.48.1, |
|00001df0| 11 34 38 11 31 29 20 20 | 20 20 20 20 20 20 20 20 |.48.1)  |        |
|00001e00| 20 20 20 20 20 20 20 20 | 44 6f 6d 61 69 6e 3a 20 |        |Domain: |
|00001e10| 28 2d 11 34 38 11 31 2c | 20 11 34 38 11 31 29 0d |(-.48.1,| .48.1).|
|00001e20| 0a 00 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |.....   |        |
|00001e30| 20 20 20 20 20 20 20 20 | 20 20 20 34 20 20 20 20 |        |   4    |
|00001e40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001e50| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 33 20 20 |        |   .23  |
|00001e60| 20 20 20 32 0d 0b 00 20 | 20 20 20 20 11 31 0e 33 |   2... |    .1.3|
|00001e70| 2d 36 2d 35 63 0e 28 63 | 29 0f 20 28 11 33 66 11 |-6-5c.(c|). (.3f.|
|00001e80| 31 2f 11 33 67 11 31 29 | 28 11 33 78 11 31 29 20 |1/.3g.1)|(.3x.1) |
|00001e90| 3d 20 11 34 32 32 32 32 | 32 20 20 20 20 20 20 20 |= .42222|2       |
|00001ea0| 20 20 20 20 20 20 11 31 | 0e 33 2d 36 2d 35 64 0e |      .1|.3-6-5d.|
|00001eb0| 28 64 29 0f 20 28 11 33 | 66 67 11 31 29 28 11 33 |(d). (.3|fg.1)(.3|
|00001ec0| 78 11 31 29 20 3d 20 34 | 11 33 78 20 20 11 31 2d |x.1) = 4|.3x  .1-|
|00001ed0| 20 33 11 33 78 20 20 11 | 31 2d 20 33 36 11 33 78 | 3.3x  .|1- 36.3x|
|00001ee0| 20 11 31 2b 20 32 37 0d | 0b 00 20 20 20 20 20 20 | .1+ 27.|..      |
|00001ef0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |        |      .3|
|00001f00| 78 20 11 31 2d 20 33 0d | 0a 00 20 20 20 20 20 20 |x .1- 3.|..      |
|00001f10| 20 20 20 44 6f 6d 61 69 | 6e 3a 20 11 33 78 20 11 |   Domai|n: .3x .|
|00001f20| 34 3d 20 11 31 33 20 20 | 20 20 20 20 20 20 20 20 |4= .13  |        |
|00001f30| 20 20 20 20 20 20 20 20 | 20 20 44 6f 6d 61 69 6e |        |  Domain|
|00001f40| 3a 20 28 2d 11 34 38 11 | 31 2c 20 11 34 38 11 31 |: (-.48.|1, .48.1|
|00001f50| 29 0d 0a 00 0d 0a 00 59 | 6f 75 20 6d 61 79 20 77 |)......Y|ou may w|
|00001f60| 61 6e 74 20 74 6f 20 72 | 65 76 69 65 77 20 10 32 |ant to r|eview .2|
|00001f70| 2d 34 2d 32 0e 65 32 2d | 34 2d 31 0e 47 75 69 64 |-4-2.e2-|4-1.Guid|
|00001f80| 65 64 20 45 78 61 6d 70 | 6c 65 20 31 2e 0f 0d 0a |ed Examp|le 1....|
|00001f90| 00 53 65 63 74 69 6f 6e | 20 32 2e 34 20 20 54 72 |.Section| 2.4  Tr|
|00001fa0| 61 6e 73 6c 61 74 69 6f | 6e 73 20 61 6e 64 20 43 |anslatio|ns and C|
|00001fb0| 6f 6d 62 69 6e 61 74 69 | 6f 6e 73 20 20 20 20 20 |ombinati|ons     |
|00001fc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001fd0| 20 20 20 11 34 44 32 20 | 20 20 20 20 20 20 20 20 |   .4D2 |        |
|00001fe0| 20 20 20 20 11 32 32 0d | 0b 00 20 11 31 32 2e 20 |    .22.|.. .12. |
|00001ff0| 20 47 69 76 65 6e 20 74 | 68 61 74 20 11 33 66 11 | Given t|hat .3f.|
|00002000| 31 28 11 33 78 11 31 29 | 20 3d 20 11 34 53 20 11 |1(.3x.1)| = .4S .|
|00002010| 33 78 20 11 31 61 6e 64 | 20 11 33 67 11 31 28 11 |3x .1and| .3g.1(.|
|00002020| 33 78 11 31 29 20 3d 20 | 11 33 78 20 20 11 31 2b |3x.1) = |.3x  .1+|
|00002030| 20 31 2c 20 77 68 69 63 | 68 20 6f 66 20 74 68 65 | 1, whic|h of the|
|00002040| 20 66 6f 6c 6c 6f 77 69 | 6e 67 20 69 73 20 0d 0a | followi|ng is ..|
|00002050| 00 20 20 20 20 20 63 6f | 72 72 65 63 74 3f 0d 0a |.     co|rrect?..|
|00002060| 00 0d 0a 00 20 20 20 20 | 20 0e 33 2d 36 2d 39 61 |....    | .3-6-9a|
|00002070| 0e 28 61 29 0f 20 28 11 | 33 66 20 11 31 2b 20 11 |.(a). (.|3f .1+ .|
|00002080| 33 67 11 31 29 28 34 29 | 20 3d 20 32 31 20 20 20 |3g.1)(4)| = 21   |
|00002090| 20 20 20 20 20 20 20 20 | 20 20 20 0e 33 2d 36 2d |        |   .3-6-|
|000020a0| 39 62 0e 28 62 29 0f 20 | 28 11 33 66 20 11 31 2d |9b.(b). |(.3f .1-|
|000020b0| 20 11 33 67 11 31 29 28 | 39 29 20 3d 20 2d 37 37 | .3g.1)(|9) = -77|
|000020c0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 0e 33 2d 36 2d |......  |   .3-6-|
|000020d0| 39 63 0e 28 63 29 0f 20 | 28 11 33 66 20 11 31 2b |9c.(c). |(.3f .1+|
|000020e0| 20 11 33 67 11 31 29 28 | 2d 31 29 20 3d 20 31 20 | .3g.1)(|-1) = 1 |
|000020f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 0e 33 2d |        |     .3-|
|00002100| 36 2d 39 64 0e 28 64 29 | 0f 20 28 11 33 66 20 11 |6-9d.(d)|. (.3f .|
|00002110| 31 2d 20 11 33 67 11 31 | 29 28 31 29 20 3d 20 2d |1- .3g.1|)(1) = -|
|00002120| 31 0d 0a 00 0d 0a 00 59 | 6f 75 20 6d 61 79 20 77 |1......Y|ou may w|
|00002130| 61 6e 74 20 74 6f 20 72 | 65 76 69 65 77 20 10 32 |ant to r|eview .2|
|00002140| 2d 34 2d 32 0e 65 32 2d | 34 2d 32 0e 47 75 69 64 |-4-2.e2-|4-2.Guid|
|00002150| 65 64 20 45 78 61 6d 70 | 6c 65 20 32 2e 0f 0d 0a |ed Examp|le 2....|
|00002160| 00 0d 0a 00 53 65 63 74 | 69 6f 6e 20 32 2e 34 20 |....Sect|ion 2.4 |
|00002170| 20 54 72 61 6e 73 6c 61 | 74 69 6f 6e 73 20 61 6e | Transla|tions an|
|00002180| 64 20 43 6f 6d 62 69 6e | 61 74 69 6f 6e 73 20 20 |d Combin|ations  |
|00002190| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000021a0| 11 32 32 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.22     |        |
|000021b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000021c0| 20 20 20 20 20 11 33 66 | 0d 0b 00 20 11 31 33 2e |     .3f|... .13.|
|000021d0| 20 20 4c 65 74 20 11 33 | 66 11 31 28 11 33 78 11 |  Let .3|f.1(.3x.|
|000021e0| 31 29 20 3d 20 32 11 33 | 78 20 20 11 31 2d 20 31 |1) = 2.3|x  .1- 1|
|000021f0| 20 61 6e 64 20 11 33 67 | 11 31 28 11 33 78 11 31 | and .3g|.1(.3x.1|
|00002200| 29 20 3d 20 11 33 78 20 | 11 31 2b 20 32 2e 20 20 |) = .3x |.1+ 2.  |
|00002210| 45 76 61 6c 75 61 74 65 | 20 28 11 34 32 11 31 29 |Evaluate| (.42.1)|
|00002220| 28 2d 31 29 20 2d 20 11 | 33 67 11 31 28 33 29 2e |(-1) - .|3g.1(3).|
|00002230| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00002240| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002250| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002260| 20 20 20 20 20 20 20 20 | 11 33 67 0d 0a 00 0d 0a |        |.3g.....|
|00002270| 00 0d 0b 00 20 20 20 20 | 20 11 31 0e 33 2d 36 2d |....    | .1.3-6-|
|00002280| 31 39 61 0e 28 61 29 0f | 20 31 20 20 20 20 20 20 |19a.(a).| 1      |
|00002290| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000022a0| 20 20 20 20 20 20 0e 33 | 2d 36 2d 31 39 62 0e 28 |      .3|-6-19b.(|
|000022b0| 62 29 0f 20 2d 34 0d 0a | 00 0d 0b 00 20 20 20 20 |b). -4..|....    |
|000022c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000022d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000022e0| 20 20 20 20 20 20 20 34 | 0d 0b 00 20 20 20 20 20 |       4|...     |
|000022f0| 0e 33 2d 36 2d 31 39 63 | 0e 28 63 29 0f 20 2d 36 |.3-6-19c|.(c). -6|
|00002300| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002310| 20 20 20 20 20 20 20 20 | 20 20 20 0e 33 2d 36 2d |        |   .3-6-|
|00002320| 31 39 64 0e 28 64 29 0f | 20 2d 11 34 32 0d 0b 00 |19d.(d).| -.42...|
|00002330| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002340| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002350| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 33 0d 0a |        |   .13..|
|00002360| 00 0d 0a 00 59 6f 75 20 | 6d 61 79 20 77 61 6e 74 |....You |may want|
|00002370| 20 74 6f 20 72 65 76 69 | 65 77 20 10 32 2d 34 2d | to revi|ew .2-4-|
|00002380| 32 0e 65 32 2d 34 2d 32 | 0e 47 75 69 64 65 64 20 |2.e2-4-2|.Guided |
|00002390| 45 78 61 6d 70 6c 65 20 | 32 2e 0f 0d 0a 00 53 65 |Example |2.....Se|
|000023a0| 63 74 69 6f 6e 20 32 2e | 34 20 20 54 72 61 6e 73 |ction 2.|4  Trans|
|000023b0| 6c 61 74 69 6f 6e 73 20 | 61 6e 64 20 43 6f 6d 62 |lations |and Comb|
|000023c0| 69 6e 61 74 69 6f 6e 73 | 0d 0b 00 20 34 2e 20 20 |inations|... 4.  |
|000023d0| 4c 65 74 20 11 33 66 11 | 31 28 11 33 78 11 31 29 |Let .3f.|1(.3x.1)|
|000023e0| 20 3d 20 32 11 33 78 20 | 11 31 2b 20 36 20 61 6e | = 2.3x |.1+ 6 an|
|000023f0| 64 20 11 33 67 11 31 28 | 11 33 78 11 31 29 20 3d |d .3g.1(|.3x.1) =|
|00002400| 20 36 20 2d 20 11 33 78 | 11 31 2e 20 20 57 68 69 | 6 - .3x|.1.  Whi|
|00002410| 63 68 20 6f 66 20 74 68 | 65 20 66 6f 6c 6c 6f 77 |ch of th|e follow|
|00002420| 69 6e 67 20 69 73 20 11 | 33 69 6e 63 6f 72 72 65 |ing is .|3incorre|
|00002430| 63 74 11 31 3f 0d 0a 00 | 0d 0a 00 20 20 20 20 20 |ct.1?...|...     |
|00002440| 0e 33 2d 36 2d 32 33 61 | 0e 28 61 29 0f 20 28 11 |.3-6-23a|.(a). (.|
|00002450| 33 66 20 11 34 6f 20 11 | 33 67 11 31 29 28 11 33 |3f .4o .|3g.1)(.3|
|00002460| 78 11 31 29 20 3d 20 31 | 38 20 2d 20 32 11 33 78 |x.1) = 1|8 - 2.3x|
|00002470| 20 20 20 20 20 20 20 20 | 20 11 31 0e 33 2d 36 2d |        | .1.3-6-|
|00002480| 32 33 62 0e 28 62 29 0f | 20 28 11 33 67 20 11 34 |23b.(b).| (.3g .4|
|00002490| 6f 20 11 33 66 11 31 29 | 28 11 33 78 11 31 29 20 |o .3f.1)|(.3x.1) |
|000024a0| 3d 20 2d 32 11 33 78 0d | 0a 00 0d 0b 00 20 20 20 |= -2.3x.|.....   |
|000024b0| 20 20 11 31 0e 33 2d 36 | 2d 32 33 63 0e 28 63 29 |  .1.3-6|-23c.(c)|
|000024c0| 0f 20 28 11 33 66 20 11 | 34 6f 20 11 33 66 11 31 |. (.3f .|4o .3f.1|
|000024d0| 29 28 11 33 78 11 31 29 | 20 3d 20 34 11 33 78 20 |)(.3x.1)| = 4.3x |
|000024e0| 11 31 2b 20 31 38 20 20 | 20 20 20 20 20 20 20 0e |.1+ 18  |       .|
|000024f0| 33 2d 36 2d 32 33 64 0e | 28 64 29 0f 20 41 6c 6c |3-6-23d.|(d). All|
|00002500| 20 61 72 65 20 63 6f 72 | 72 65 63 74 0d 0a 00 0d | are cor|rect....|
|00002510| 0a 00 59 6f 75 20 6d 61 | 79 20 77 61 6e 74 20 74 |..You ma|y want t|
|00002520| 6f 20 72 65 76 69 65 77 | 20 10 32 2d 34 2d 32 0e |o review| .2-4-2.|
|00002530| 65 32 2d 34 2d 33 0e 47 | 75 69 64 65 64 20 45 78 |e2-4-3.G|uided Ex|
|00002540| 61 6d 70 6c 65 20 33 2e | 0f 0d 0a 00 53 65 63 74 |ample 3.|....Sect|
|00002550| 69 6f 6e 20 32 2e 34 20 | 20 54 72 61 6e 73 6c 61 |ion 2.4 | Transla|
|00002560| 74 69 6f 6e 73 20 61 6e | 64 20 43 6f 6d 62 69 6e |tions an|d Combin|
|00002570| 61 74 69 6f 6e 73 20 20 | 20 20 20 20 20 20 20 20 |ations  |        |
|00002580| 20 20 20 20 20 20 20 20 | 11 34 67 32 32 32 32 0d |        |.4g2222.|
|00002590| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000025a0| 20 20 20 66 20 20 20 20 | 20 20 20 20 20 20 20 20 |   f    |        |
|000025b0| 20 20 20 20 20 20 11 32 | 32 0d 0b 00 20 11 31 35 |      .2|2... .15|
|000025c0| 2e 20 20 4c 65 74 20 11 | 33 66 11 31 28 11 33 78 |.  Let .|3f.1(.3x|
|000025d0| 11 31 29 20 3d 20 11 34 | 76 20 11 33 78 20 11 31 |.1) = .4|v .3x .1|
|000025e0| 2b 20 33 20 61 6e 64 20 | 11 33 67 11 31 28 11 33 |+ 3 and |.3g.1(.3|
|000025f0| 78 11 31 29 20 3d 20 11 | 33 78 20 20 11 31 2d 20 |x.1) = .|3x  .1- |
|00002600| 31 2e 20 20 46 69 6e 64 | 20 11 33 66 20 11 34 6f |1.  Find| .3f .4o|
|00002610| 20 11 33 67 20 11 31 61 | 6e 64 20 11 33 67 20 11 | .3g .1a|nd .3g .|
|00002620| 34 6f 20 11 33 66 11 31 | 2e 0d 0a 00 0d 0a 00 20 |4o .3f.1|....... |
|00002630| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002640| 20 20 20 20 20 20 20 11 | 34 67 32 32 32 32 32 0d |       .|4g22222.|
|00002650| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00002660| 20 20 20 20 20 20 20 20 | 20 66 20 11 32 32 0d 0b |        | f .22..|
|00002670| 00 20 20 20 20 20 11 31 | 0e 33 2d 36 2d 32 35 61 |.     .1|.3-6-25a|
|00002680| 0e 28 61 29 0f 20 28 11 | 33 66 20 11 34 6f 20 11 |.(a). (.|3f .4o .|
|00002690| 33 67 11 31 29 28 11 33 | 78 11 31 29 20 3d 20 11 |3g.1)(.3|x.1) = .|
|000026a0| 34 76 20 11 33 78 20 11 | 31 2b 20 32 20 20 20 20 |4v .3x .|1+ 2    |
|000026b0| 20 20 20 20 20 20 20 20 | 28 11 33 67 20 11 34 6f |        |(.3g .4o|
|000026c0| 20 11 33 66 11 31 29 28 | 11 33 78 11 31 29 20 3d | .3f.1)(|.3x.1) =|
|000026d0| 20 11 33 78 20 11 31 2b | 20 32 0d 0a 00 0d 0b 00 | .3x .1+| 2......|
|000026e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000026f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002700| 11 34 67 32 32 32 32 20 | 20 20 20 20 20 20 20 20 |.4g2222 |        |
|00002710| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002720| 20 20 67 32 32 32 32 0d | 0b 00 20 20 20 20 20 20 |  g2222.|..      |
|00002730| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002740| 20 20 11 32 32 20 20 20 | 20 20 20 11 34 66 20 20 |  .22   |   .4f  |
|00002750| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002760| 20 20 20 20 20 20 11 32 | 32 20 20 20 20 20 20 11 |      .2|2      .|
|00002770| 34 66 0d 0b 00 20 20 20 | 20 20 11 31 0e 33 2d 36 |4f...   |  .1.3-6|
|00002780| 2d 32 35 62 0e 28 62 29 | 0f 20 28 11 33 66 20 11 |-25b.(b)|. (.3f .|
|00002790| 34 6f 20 11 33 67 11 31 | 29 28 11 33 78 11 31 29 |4o .3g.1|)(.3x.1)|
|000027a0| 20 3d 20 28 11 33 78 20 | 20 11 31 2d 20 31 29 11 | = (.3x | .1- 1).|
|000027b0| 34 76 20 11 33 78 20 11 | 31 2b 20 33 20 20 20 20 |4v .3x .|1+ 3    |
|000027c0| 28 11 33 67 20 11 34 6f | 20 11 33 66 11 31 29 28 |(.3g .4o| .3f.1)(|
|000027d0| 11 33 78 11 31 29 20 3d | 20 28 11 33 78 20 20 11 |.3x.1) =| (.3x  .|
|000027e0| 31 2d 20 31 29 11 34 76 | 20 11 33 78 20 11 31 2b |1- 1).4v| .3x .1+|
|000027f0| 20 33 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 | 3......|        |
|00002800| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002810| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002820| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002830| 11 34 67 32 32 32 32 32 | 0d 0b 00 20 20 20 20 20 |.4g22222|...     |
|00002840| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002850| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002860| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002870| 20 20 66 20 11 32 32 0d | 0b 00 20 20 20 20 20 11 |  f .22.|..     .|
|00002880| 31 0e 33 2d 36 2d 32 35 | 63 0e 28 63 29 0f 20 28 |1.3-6-25|c.(c). (|
|00002890| 11 33 66 20 11 34 6f 20 | 11 33 67 11 31 29 28 11 |.3f .4o |.3g.1)(.|
|000028a0| 33 78 11 31 29 20 3d 20 | 11 33 78 20 11 31 2b 20 |3x.1) = |.3x .1+ |
|000028b0| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 28 |2       |       (|
|000028c0| 11 33 67 20 11 34 6f 20 | 11 33 66 11 31 29 28 11 |.3g .4o |.3f.1)(.|
|000028d0| 33 78 11 31 29 20 3d 20 | 11 34 76 20 11 33 78 20 |3x.1) = |.4v .3x |
|000028e0| 20 11 31 2b 20 32 0d 0a | 00 0d 0a 00 0d 0b 00 20 | .1+ 2..|....... |
|000028f0| 20 20 20 20 0e 33 2d 36 | 2d 32 35 64 0e 28 64 29 |    .3-6|-25d.(d)|
|00002900| 0f 20 28 11 33 66 20 11 | 34 6f 20 11 33 67 11 31 |. (.3f .|4o .3g.1|
|00002910| 29 28 11 33 78 11 31 29 | 20 3d 20 11 33 78 20 11 |)(.3x.1)| = .3x .|
|00002920| 31 2b 20 32 20 20 20 20 | 20 20 20 20 20 20 20 20 |1+ 2    |        |
|00002930| 20 20 28 11 33 67 20 11 | 34 6f 20 11 33 66 11 31 |  (.3g .|4o .3f.1|
|00002940| 29 28 11 33 78 11 31 29 | 20 3d 20 11 33 78 20 11 |)(.3x.1)| = .3x .|
|00002950| 31 2b 20 32 0d 0a 00 0d | 0a 00 0d 0b 00 59 6f 75 |1+ 2....|.....You|
|00002960| 20 6d 61 79 20 77 61 6e | 74 20 74 6f 20 72 65 76 | may wan|t to rev|
|00002970| 69 65 77 20 10 32 2d 34 | 2d 32 0e 65 32 2d 34 2d |iew .2-4|-2.e2-4-|
|00002980| 34 0e 47 75 69 64 65 64 | 20 45 78 61 6d 70 6c 65 |4.Guided| Example|
|00002990| 20 34 2e 0f 0d 0a 00 53 | 65 63 74 69 6f 6e 20 32 | 4.....S|ection 2|
|000029a0| 2e 34 20 20 54 72 61 6e | 73 6c 61 74 69 6f 6e 73 |.4  Tran|slations|
|000029b0| 20 61 6e 64 20 43 6f 6d | 62 69 6e 61 74 69 6f 6e | and Com|bination|
|000029c0| 73 0d 0b 00 20 36 2e 20 | 20 4c 65 74 20 11 33 66 |s... 6. | Let .3f|
|000029d0| 11 31 28 11 33 78 11 31 | 29 20 3d 20 7c 11 33 78 |.1(.3x.1|) = |.3x|
|000029e0| 20 11 31 2d 20 34 7c 20 | 61 6e 64 20 11 33 67 11 | .1- 4| |and .3g.|
|000029f0| 31 28 11 33 78 11 31 29 | 20 3d 20 11 33 78 20 11 |1(.3x.1)| = .3x .|
|00002a00| 31 2b 20 32 2e 20 20 46 | 69 6e 64 20 11 33 66 20 |1+ 2.  F|ind .3f |
|00002a10| 11 34 6f 20 11 33 67 20 | 11 31 61 6e 64 20 11 33 |.4o .3g |.1and .3|
|00002a20| 67 20 11 34 6f 20 11 33 | 66 11 31 2e 0d 0a 00 0d |g .4o .3|f.1.....|
|00002a30| 0a 00 20 20 20 20 20 0e | 33 2d 36 2d 33 31 61 0e |..     .|3-6-31a.|
|00002a40| 28 61 29 0f 20 28 11 33 | 66 20 11 34 6f 20 11 33 |(a). (.3|f .4o .3|
|00002a50| 67 11 31 29 28 11 33 78 | 11 31 29 20 3d 20 7c 11 |g.1)(.3x|.1) = |.|
|00002a60| 33 78 20 11 31 2d 20 34 | 7c 28 11 33 78 20 11 31 |3x .1- 4||(.3x .1|
|00002a70| 2b 20 32 29 20 20 20 28 | 11 33 67 20 11 34 6f 20 |+ 2)   (|.3g .4o |
|00002a80| 11 33 66 11 31 29 28 11 | 33 78 11 31 29 20 3d 20 |.3f.1)(.|3x.1) = |
|00002a90| 7c 11 33 78 20 11 31 2d | 20 34 7c 28 11 33 78 20 ||.3x .1-| 4|(.3x |
|00002aa0| 11 31 2b 20 32 29 0d 0a | 00 0d 0a 00 20 20 20 20 |.1+ 2)..|....    |
|00002ab0| 20 0e 33 2d 36 2d 33 31 | 62 0e 28 62 29 0f 20 28 | .3-6-31|b.(b). (|
|00002ac0| 11 33 66 20 11 34 6f 20 | 11 33 67 11 31 29 28 11 |.3f .4o |.3g.1)(.|
|00002ad0| 33 78 11 31 29 20 3d 20 | 7c 11 33 78 20 11 31 2d |3x.1) = ||.3x .1-|
|00002ae0| 20 32 7c 20 20 20 20 20 | 20 20 20 20 20 28 11 33 | 2|     |     (.3|
|00002af0| 67 20 11 34 6f 20 11 33 | 66 11 31 29 28 11 33 78 |g .4o .3|f.1)(.3x|
|00002b00| 11 31 29 20 3d 20 7c 11 | 33 78 20 11 31 2d 20 34 |.1) = |.|3x .1- 4|
|00002b10| 7c 20 2b 20 32 0d 0a 00 | 0d 0a 00 20 20 20 20 20 || + 2...|...     |
|00002b20| 0e 33 2d 36 2d 33 31 63 | 0e 28 63 29 0f 20 28 11 |.3-6-31c|.(c). (.|
|00002b30| 33 66 20 11 34 6f 20 11 | 33 67 11 31 29 28 11 33 |3f .4o .|3g.1)(.3|
|00002b40| 78 11 31 29 20 3d 20 7c | 11 33 78 20 11 31 2d 20 |x.1) = ||.3x .1- |
|00002b50| 34 7c 20 2b 20 32 20 20 | 20 20 20 20 28 11 33 67 |4| + 2  |    (.3g|
|00002b60| 20 11 34 6f 20 11 33 66 | 11 31 29 28 11 33 78 11 | .4o .3f|.1)(.3x.|
|00002b70| 31 29 20 3d 20 7c 11 33 | 78 20 11 31 2d 20 32 7c |1) = |.3|x .1- 2||
|00002b80| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|00002b90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00002ba0| 32 0d 0b 00 20 20 20 20 | 20 11 31 0e 33 2d 36 2d |2...    | .1.3-6-|
|00002bb0| 33 31 64 0e 28 64 29 0f | 20 28 11 33 66 20 11 34 |31d.(d).| (.3f .4|
|00002bc0| 6f 20 11 33 67 11 31 29 | 28 11 33 78 11 31 29 20 |o .3g.1)|(.3x.1) |
|00002bd0| 3d 20 7c 11 33 78 20 20 | 11 31 2d 20 32 11 33 78 |= |.3x  |.1- 2.3x|
|00002be0| 20 11 31 2d 20 38 7c 20 | 20 20 20 28 11 33 66 20 | .1- 8| |   (.3f |
|00002bf0| 11 34 6f 20 11 33 67 11 | 31 29 28 11 33 78 11 31 |.4o .3g.|1)(.3x.1|
|00002c00| 29 20 3d 20 7c 32 11 33 | 78 20 11 31 2d 20 32 7c |) = |2.3|x .1- 2||
|00002c10| 0d 0a 00 0d 0a 00 59 6f | 75 20 6d 61 79 20 77 61 |......Yo|u may wa|
|00002c20| 6e 74 20 74 6f 20 72 65 | 76 69 65 77 20 10 32 2d |nt to re|view .2-|
|00002c30| 34 2d 32 0e 65 32 2d 34 | 2d 35 0e 47 75 69 64 65 |4-2.e2-4|-5.Guide|
|00002c40| 64 20 45 78 61 6d 70 6c | 65 20 35 2e 0f 0d 0a 00 |d Exampl|e 5.....|
|00002c50| 53 65 63 74 69 6f 6e 20 | 32 2e 34 20 20 54 72 61 |Section |2.4  Tra|
|00002c60| 6e 73 6c 61 74 69 6f 6e | 73 20 61 6e 64 20 43 6f |nslation|s and Co|
|00002c70| 6d 62 69 6e 61 74 69 6f | 6e 73 20 20 20 20 20 20 |mbinatio|ns      |
|00002c80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002c90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002ca0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002cb0| 20 20 20 20 20 20 20 20 | 20 11 32 32 0d 0b 00 20 |        | .22... |
|00002cc0| 11 31 37 2e 20 20 46 69 | 6e 64 20 74 77 6f 20 66 |.17.  Fi|nd two f|
|00002cd0| 75 6e 63 74 69 6f 6e 73 | 20 11 33 66 20 11 31 61 |unctions| .3f .1a|
|00002ce0| 6e 64 20 11 33 67 20 11 | 31 73 75 63 68 20 74 68 |nd .3g .|1such th|
|00002cf0| 61 74 20 28 11 33 66 20 | 11 34 6f 20 11 33 67 11 |at (.3f |.4o .3g.|
|00002d00| 31 29 28 11 33 78 11 31 | 29 20 3d 20 28 11 33 78 |1)(.3x.1|) = (.3x|
|00002d10| 20 11 31 2b 20 31 32 29 | 20 2e 0d 0a 00 0d 0b 00 | .1+ 12)| .......|
|00002d20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002d30| 20 11 32 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | .22    |        |
|00002d40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002d50| 20 20 20 20 20 20 20 20 | 20 20 20 32 0d 0b 00 20 |        |   2... |
|00002d60| 20 20 20 20 11 31 0e 61 | 33 2d 36 2d 33 37 61 0e |    .1.a|3-6-37a.|
|00002d70| 28 61 29 0f 20 11 33 66 | 11 31 28 11 33 78 11 31 |(a). .3f|.1(.3x.1|
|00002d80| 29 20 3d 20 11 33 78 20 | 11 31 2c 20 11 33 67 11 |) = .3x |.1, .3g.|
|00002d90| 31 28 11 33 78 11 31 29 | 20 3d 20 11 33 78 20 11 |1(.3x.1)| = .3x .|
|00002da0| 31 2b 20 31 32 20 20 20 | 20 20 20 0e 61 33 2d 36 |1+ 12   |   .a3-6|
|00002db0| 2d 33 37 62 0e 28 62 29 | 0f 20 11 33 66 11 31 28 |-37b.(b)|. .3f.1(|
|00002dc0| 11 33 78 11 31 29 20 3d | 20 28 11 33 78 20 11 31 |.3x.1) =| (.3x .1|
|00002dd0| 2b 20 31 29 20 2c 20 11 | 33 67 11 31 28 11 33 78 |+ 1) , .|3g.1(.3x|
|00002de0| 11 31 29 20 3d 20 11 33 | 78 20 11 31 2b 20 31 31 |.1) = .3|x .1+ 11|
|00002df0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00002e00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002e10| 20 20 20 20 20 11 32 32 | 0d 0b 00 20 20 20 20 20 |     .22|...     |
|00002e20| 11 31 0e 61 33 2d 36 2d | 33 37 63 0e 28 63 29 0f |.1.a3-6-|37c.(c).|
|00002e30| 20 11 33 66 11 31 28 11 | 33 78 11 31 29 20 3d 20 | .3f.1(.|3x.1) = |
|00002e40| 11 33 78 11 31 2c 20 11 | 33 67 11 31 28 11 33 78 |.3x.1, .|3g.1(.3x|
|00002e50| 11 31 29 20 3d 20 28 11 | 33 78 20 11 31 2b 20 31 |.1) = (.|3x .1+ 1|
|00002e60| 32 29 20 20 20 20 20 0e | 61 33 2d 36 2d 33 37 64 |2)     .|a3-6-37d|
|00002e70| 0e 28 64 29 0f 20 41 6c | 6c 20 61 72 65 20 63 6f |.(d). Al|l are co|
|00002e80| 72 72 65 63 74 0d 0a 00 | 0d 0a 00 59 6f 75 20 6d |rrect...|...You m|
|00002e90| 61 79 20 77 61 6e 74 20 | 74 6f 20 72 65 76 69 65 |ay want |to revie|
|00002ea0| 77 20 10 32 2d 34 2d 32 | 0e 69 32 2d 34 2d 31 0e |w .2-4-2|.i2-4-1.|
|00002eb0| 49 6e 74 65 67 72 61 74 | 65 64 20 45 78 61 6d 70 |Integrat|ed Examp|
|00002ec0| 6c 65 20 31 2e 0f 0d 0a | 00 53 65 63 74 69 6f 6e |le 1....|.Section|
|00002ed0| 20 32 2e 34 20 20 54 72 | 61 6e 73 6c 61 74 69 6f | 2.4  Tr|anslatio|
|00002ee0| 6e 73 20 61 6e 64 20 43 | 6f 6d 62 69 6e 61 74 69 |ns and C|ombinati|
|00002ef0| 6f 6e 73 0d 0b 00 20 38 | 2e 20 20 46 69 6e 64 20 |ons... 8|.  Find |
|00002f00| 74 77 6f 20 66 75 6e 63 | 74 69 6f 6e 73 20 11 33 |two func|tions .3|
|00002f10| 66 20 11 31 61 6e 64 20 | 11 33 67 20 11 31 73 75 |f .1and |.3g .1su|
|00002f20| 63 68 20 74 68 61 74 20 | 28 11 33 66 20 11 34 6f |ch that |(.3f .4o|
|00002f30| 20 11 33 67 11 31 29 28 | 11 33 78 11 31 29 20 3d | .3g.1)(|.3x.1) =|
|00002f40| 20 31 2f 28 33 11 33 78 | 20 11 31 2d 20 31 29 2e | 1/(3.3x| .1- 1).|
|00002f50| 0d 0a 00 0d 0a 00 20 20 | 20 20 20 0e 61 33 2d 36 |......  |   .a3-6|
|00002f60| 2d 34 31 61 0e 28 61 29 | 0f 20 11 33 66 11 31 28 |-41a.(a)|. .3f.1(|
|00002f70| 11 33 78 11 31 29 20 3d | 20 31 2f 11 33 78 11 31 |.3x.1) =| 1/.3x.1|
|00002f80| 2c 20 11 33 67 11 31 28 | 11 33 78 11 31 29 20 3d |, .3g.1(|.3x.1) =|
|00002f90| 20 33 11 33 78 20 11 31 | 2d 20 31 20 20 20 20 20 | 3.3x .1|- 1     |
|00002fa0| 20 0e 61 33 2d 36 2d 34 | 31 62 0e 28 62 29 0f 20 | .a3-6-4|1b.(b). |
|00002fb0| 11 33 66 11 31 28 11 33 | 78 11 31 29 20 3d 20 33 |.3f.1(.3|x.1) = 3|
|00002fc0| 11 33 78 20 11 31 2d 20 | 31 2c 20 11 33 67 11 31 |.3x .1- |1, .3g.1|
|00002fd0| 28 11 33 78 11 31 29 20 | 3d 20 31 2f 11 33 78 0d |(.3x.1) |= 1/.3x.|
|00002fe0| 0a 00 0d 0b 00 20 20 20 | 20 20 11 31 0e 61 33 2d |.....   |  .1.a3-|
|00002ff0| 36 2d 34 31 63 0e 28 63 | 29 0f 20 11 33 66 11 31 |6-41c.(c|). .3f.1|
|00003000| 28 11 33 78 11 31 29 20 | 3d 20 31 2f 11 33 78 11 |(.3x.1) |= 1/.3x.|
|00003010| 31 2c 20 11 33 67 11 31 | 28 11 33 78 11 31 29 20 |1, .3g.1|(.3x.1) |
|00003020| 3d 20 31 2f 28 33 11 33 | 78 20 11 31 2d 20 31 29 |= 1/(3.3|x .1- 1)|
|00003030| 20 20 0e 61 33 2d 36 2d | 34 31 64 0e 28 64 29 0f |  .a3-6-|41d.(d).|
|00003040| 20 41 6c 6c 20 61 72 65 | 20 63 6f 72 72 65 63 74 | All are| correct|
|00003050| 0d 0a 00 0d 0a 00 59 6f | 75 20 6d 61 79 20 77 61 |......Yo|u may wa|
|00003060| 6e 74 20 74 6f 20 72 65 | 76 69 65 77 20 10 32 2d |nt to re|view .2-|
|00003070| 34 2d 32 0e 69 32 2d 34 | 2d 31 0e 49 6e 74 65 67 |4-2.i2-4|-1.Integ|
|00003080| 72 61 74 65 64 20 45 78 | 61 6d 70 6c 65 20 31 2e |rated Ex|ample 1.|
|00003090| 0f 0d 0a 00 53 65 63 74 | 69 6f 6e 20 32 2e 34 20 |....Sect|ion 2.4 |
|000030a0| 20 54 72 61 6e 73 6c 61 | 74 69 6f 6e 73 20 61 6e | Transla|tions an|
|000030b0| 64 20 43 6f 6d 62 69 6e | 61 74 69 6f 6e 73 0d 0b |d Combin|ations..|
|000030c0| 00 20 39 2e 20 20 44 65 | 74 65 72 6d 69 6e 65 20 |. 9.  De|termine |
|000030d0| 74 68 65 20 64 6f 6d 61 | 69 6e 20 6f 66 20 11 33 |the doma|in of .3|
|000030e0| 66 20 11 34 6f 20 11 33 | 67 20 11 31 66 6f 72 20 |f .4o .3|g .1for |
|000030f0| 74 68 65 20 66 6f 6c 6c | 6f 77 69 6e 67 20 66 75 |the foll|owing fu|
|00003100| 6e 63 74 69 6f 6e 73 2e | 0d 0a 00 20 20 20 20 20 |nctions.|...     |
|00003110| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003120| 35 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 11 |5...    |       .|
|00003130| 33 66 11 31 28 11 33 78 | 11 31 29 20 3d 20 11 34 |3f.1(.3x|.1) = .4|
|00003140| 32 32 32 32 32 32 11 31 | 2c 20 20 20 20 11 33 67 |222222.1|,    .3g|
|00003150| 11 31 28 11 33 78 11 31 | 29 20 3d 20 11 33 78 20 |.1(.3x.1|) = .3x |
|00003160| 11 31 2b 20 32 0d 0b 00 | 20 20 20 20 20 20 20 20 |.1+ 2...|        |
|00003170| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|00003180| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00003190| 20 20 20 11 33 78 20 20 | 11 31 2d 20 34 0d 0a 00 |   .3x  |.1- 4...|
|000031a0| 0d 0a 00 20 20 20 20 20 | 0e 33 2d 36 2d 34 37 61 |...     |.3-6-47a|
|000031b0| 0e 28 61 29 0f 20 41 6c | 6c 20 72 65 61 6c 20 6e |.(a). Al|l real n|
|000031c0| 75 6d 62 65 72 73 20 65 | 78 63 65 70 74 20 11 33 |umbers e|xcept .3|
|000031d0| 78 20 11 31 3d 20 11 34 | 2b 11 31 32 0d 0a 00 0d |x .1= .4|+.12....|
|000031e0| 0b 00 20 20 20 20 20 0e | 33 2d 36 2d 34 37 62 0e |..     .|3-6-47b.|
|000031f0| 28 62 29 0f 20 41 6c 6c | 20 72 65 61 6c 20 6e 75 |(b). All| real nu|
|00003200| 6d 62 65 72 73 20 65 78 | 63 65 70 74 20 11 33 78 |mbers ex|cept .3x|
|00003210| 20 11 31 3d 20 2d 34 2c | 20 30 0d 0a 00 0d 0b 00 | .1= -4,| 0......|
|00003220| 20 20 20 20 20 0e 33 2d | 36 2d 34 37 63 0e 28 63 |     .3-|6-47c.(c|
|00003230| 29 0f 20 41 6c 6c 20 72 | 65 61 6c 20 6e 75 6d 62 |). All r|eal numb|
|00003240| 65 72 73 20 65 78 63 65 | 70 74 20 11 33 78 20 11 |ers exce|pt .3x .|
|00003250| 31 3d 20 30 2c 20 34 0d | 0a 00 0d 0b 00 20 20 20 |1= 0, 4.|.....   |
|00003260| 20 20 0e 33 2d 36 2d 34 | 37 64 0e 28 64 29 0f 20 |  .3-6-4|7d.(d). |
|00003270| 41 6c 6c 20 72 65 61 6c | 20 6e 75 6d 62 65 72 73 |All real| numbers|
|00003280| 0d 0a 00 0d 0a 00 59 6f | 75 20 6d 61 79 20 77 61 |......Yo|u may wa|
|00003290| 6e 74 20 74 6f 20 72 65 | 76 69 65 77 20 10 32 2d |nt to re|view .2-|
|000032a0| 34 2d 32 0e 69 32 2d 34 | 2d 32 0e 49 6e 74 65 67 |4-2.i2-4|-2.Integ|
|000032b0| 72 61 74 65 64 20 45 78 | 61 6d 70 6c 65 20 32 2e |rated Ex|ample 2.|
|000032c0| 0f 0d 0a 00 14 00 00 00 | 6b 00 00 00 50 00 04 00 |........|k...P...|
|000032d0| 14 00 00 00 10 00 00 00 | 33 2d 36 2d 31 39 61 00 |........|3-6-19a.|
|000032e0| 83 00 00 00 2d 00 00 00 | 50 00 04 00 83 00 00 00 |....-...|P.......|
|000032f0| 7f 00 00 00 33 2d 36 2d | 31 39 62 00 b4 00 00 00 |....3-6-|19b.....|
|00003300| 29 01 00 00 50 00 04 00 | b4 00 00 00 b0 00 00 00 |)...P...|........|
|00003310| 33 2d 36 2d 31 39 63 00 | e1 01 00 00 29 01 00 00 |3-6-19c.|....)...|
|00003320| 50 00 04 00 e1 01 00 00 | dd 01 00 00 33 2d 36 2d |P.......|....3-6-|
|00003330| 31 39 64 00 0e 03 00 00 | 52 00 00 00 50 00 04 00 |19d.....|R...P...|
|00003340| 0e 03 00 00 0a 03 00 00 | 33 2d 36 2d 32 33 61 00 |........|3-6-23a.|
|00003350| 64 03 00 00 52 00 00 00 | 50 00 04 00 64 03 00 00 |d...R...|P...d...|
|00003360| 60 03 00 00 33 2d 36 2d | 32 33 62 00 ba 03 00 00 |`...3-6-|23b.....|
|00003370| 52 00 00 00 50 00 04 00 | ba 03 00 00 b6 03 00 00 |R...P...|........|
|00003380| 33 2d 36 2d 32 33 63 00 | 10 04 00 00 2c 00 00 00 |3-6-23c.|....,...|
|00003390| 50 00 04 00 10 04 00 00 | 0c 04 00 00 33 2d 36 2d |P.......|....3-6-|
|000033a0| 32 33 64 00 40 04 00 00 | 42 00 00 00 50 00 04 00 |23d.@...|B...P...|
|000033b0| 40 04 00 00 3c 04 00 00 | 33 2d 36 2d 32 35 61 00 |@...<...|3-6-25a.|
|000033c0| 86 04 00 00 7e 00 00 00 | 50 00 04 00 86 04 00 00 |....~...|P.......|
|000033d0| 82 04 00 00 33 2d 36 2d | 32 35 62 00 08 05 00 00 |....3-6-|25b.....|
|000033e0| 57 00 00 00 50 00 04 00 | 08 05 00 00 04 05 00 00 |W...P...|........|
|000033f0| 33 2d 36 2d 32 35 63 00 | 63 05 00 00 80 01 00 00 |3-6-25c.|c.......|
|00003400| 50 00 04 00 63 05 00 00 | 5f 05 00 00 33 2d 36 2d |P...c...|_...3-6-|
|00003410| 32 35 64 00 e7 06 00 00 | 6e 00 00 00 50 00 04 00 |25d.....|n...P...|
|00003420| e7 06 00 00 e3 06 00 00 | 33 2d 36 2d 33 31 61 00 |........|3-6-31a.|
|00003430| 59 07 00 00 42 00 00 00 | 50 00 04 00 59 07 00 00 |Y...B...|P...Y...|
|00003440| 55 07 00 00 33 2d 36 2d | 33 31 62 00 9f 07 00 00 |U...3-6-|31b.....|
|00003450| 57 00 00 00 50 00 04 00 | 9f 07 00 00 9b 07 00 00 |W...P...|........|
|00003460| 33 2d 36 2d 33 31 63 00 | fa 07 00 00 22 01 00 00 |3-6-31c.|...."...|
|00003470| 50 00 04 00 fa 07 00 00 | f6 07 00 00 33 2d 36 2d |P.......|....3-6-|
|00003480| 33 31 64 00 20 09 00 00 | cc 00 00 00 50 00 04 00 |31d. ...|....P...|
|00003490| 20 09 00 00 1c 09 00 00 | 33 2d 36 2d 34 37 61 00 | .......|3-6-47a.|
|000034a0| f0 09 00 00 62 00 00 00 | 50 00 04 00 f0 09 00 00 |....b...|P.......|
|000034b0| ec 09 00 00 33 2d 36 2d | 34 37 62 00 56 0a 00 00 |....3-6-|47b.V...|
|000034c0| 0f 01 00 00 50 00 04 00 | 56 0a 00 00 52 0a 00 00 |....P...|V...R...|
|000034d0| 33 2d 36 2d 34 37 63 00 | 69 0b 00 00 0f 01 00 00 |3-6-47c.|i.......|
|000034e0| 50 00 04 00 69 0b 00 00 | 65 0b 00 00 33 2d 36 2d |P...i...|e...3-6-|
|000034f0| 34 37 64 00 7c 0c 00 00 | 2a 01 00 00 50 00 04 00 |47d.|...|*...P...|
|00003500| 7c 0c 00 00 78 0c 00 00 | 33 2d 36 2d 35 61 00 aa ||...x...|3-6-5a..|
|00003510| 0d 00 00 59 01 00 00 50 | 00 04 00 aa 0d 00 00 a6 |...Y...P|........|
|00003520| 0d 00 00 33 2d 36 2d 35 | 62 00 07 0f 00 00 62 01 |...3-6-5|b.....b.|
|00003530| 00 00 50 00 04 00 07 0f | 00 00 03 0f 00 00 33 2d |..P.....|......3-|
|00003540| 36 2d 35 63 00 6d 10 00 | 00 a5 00 00 00 50 00 04 |6-5c.m..|.....P..|
|00003550| 00 6d 10 00 00 69 10 00 | 00 33 2d 36 2d 35 64 00 |.m...i..|.3-6-5d.|
|00003560| 16 11 00 00 8c 01 00 00 | 50 00 04 00 16 11 00 00 |........|P.......|
|00003570| 12 11 00 00 33 2d 36 2d | 39 61 00 a6 12 00 00 92 |....3-6-|9a......|
|00003580| 01 00 00 50 00 04 00 a6 | 12 00 00 a2 12 00 00 33 |...P....|.......3|
|00003590| 2d 36 2d 39 62 00 3c 14 | 00 00 86 01 00 00 50 00 |-6-9b.<.|......P.|
|000035a0| 04 00 3c 14 00 00 38 14 | 00 00 33 2d 36 2d 39 63 |..<...8.|..3-6-9c|
|000035b0| 00 c6 15 00 00 8c 00 00 | 00 50 00 04 00 c6 15 00 |........|.P......|
|000035c0| 00 c2 15 00 00 33 2d 36 | 2d 39 64 00 56 16 00 00 |.....3-6|-9d.V...|
|000035d0| 58 00 00 00 50 00 04 00 | 56 16 00 00 52 16 00 00 |X...P...|V...R...|
|000035e0| 61 33 2d 36 2d 33 37 61 | 00 b2 16 00 00 58 00 00 |a3-6-37a|.....X..|
|000035f0| 00 50 00 04 00 b2 16 00 | 00 ae 16 00 00 61 33 2d |.P......|.....a3-|
|00003600| 36 2d 33 37 62 00 0e 17 | 00 00 58 00 00 00 50 00 |6-37b...|..X...P.|
|00003610| 04 00 0e 17 00 00 0a 17 | 00 00 61 33 2d 36 2d 33 |........|..a3-6-3|
|00003620| 37 63 00 6a 17 00 00 45 | 00 00 00 50 00 04 00 6a |7c.j...E|...P...j|
|00003630| 17 00 00 66 17 00 00 61 | 33 2d 36 2d 33 37 64 00 |...f...a|3-6-37d.|
|00003640| b3 17 00 00 fd 00 00 00 | 50 00 04 00 b3 17 00 00 |........|P.......|
|00003650| af 17 00 00 61 33 2d 36 | 2d 34 31 61 00 b4 18 00 |....a3-6|-41a....|
|00003660| 00 27 01 00 00 50 00 04 | 00 b4 18 00 00 b0 18 00 |.'...P..|........|
|00003670| 00 61 33 2d 36 2d 34 31 | 62 00 df 19 00 00 a4 00 |.a3-6-41|b.......|
|00003680| 00 00 50 00 04 00 df 19 | 00 00 db 19 00 00 61 33 |..P.....|......a3|
|00003690| 2d 36 2d 34 31 63 00 87 | 1a 00 00 34 00 00 00 50 |-6-41c..|...4...P|
|000036a0| 00 04 00 87 1a 00 00 83 | 1a 00 00 61 33 2d 36 2d |........|...a3-6-|
|000036b0| 34 31 64 00 e5 1a 00 00 | 56 01 00 00 4d 2a 00 00 |41d.....|V...M*..|
|000036c0| bb 1a 00 00 00 00 00 00 | 78 32 2d 34 00 65 1c 00 |........|x2-4.e..|
|000036d0| 00 2c 03 00 00 4d 2a 00 | 00 3b 1c 00 00 00 00 00 |.,...M*.|.;......|
|000036e0| 00 78 32 2d 34 2d 31 00 | bb 1f 00 00 a9 01 00 00 |.x2-4-1.|........|
|000036f0| 4d 2a 00 00 91 1f 00 00 | 00 00 00 00 78 32 2d 34 |M*......|....x2-4|
|00003700| 2d 32 00 8e 21 00 00 10 | 02 00 00 4d 2a 00 00 64 |-2..!...|...M*..d|
|00003710| 21 00 00 00 00 00 00 78 | 32 2d 34 2d 33 00 c8 23 |!......x|2-4-3..#|
|00003720| 00 00 84 01 00 00 4d 2a | 00 00 9e 23 00 00 00 00 |......M*|...#....|
|00003730| 00 00 78 32 2d 34 2d 34 | 00 76 25 00 00 21 04 00 |..x2-4-4|.v%..!..|
|00003740| 00 4d 2a 00 00 4c 25 00 | 00 00 00 00 00 78 32 2d |.M*..L%.|.....x2-|
|00003750| 34 2d 35 00 c1 29 00 00 | 8f 02 00 00 4d 2a 00 00 |4-5..)..|....M*..|
|00003760| 97 29 00 00 00 00 00 00 | 78 32 2d 34 2d 36 00 7a |.)......|x2-4-6.z|
|00003770| 2c 00 00 4f 02 00 00 4d | 2a 00 00 50 2c 00 00 00 |,..O...M|*..P,...|
|00003780| 00 00 00 78 32 2d 34 2d | 37 00 f3 2e 00 00 a1 01 |...x2-4-|7.......|
|00003790| 00 00 4d 2a 00 00 c9 2e | 00 00 00 00 00 00 78 32 |..M*....|......x2|
|000037a0| 2d 34 2d 38 00 be 30 00 | 00 06 02 00 00 4d 2a 00 |-4-8..0.|.....M*.|
|000037b0| 00 94 30 00 00 00 00 00 | 00 78 32 2d 34 2d 39 00 |..0.....|.x2-4-9.|
+--------+-------------------------+-------------------------+--------+--------+