Interactive Algebra & Trigonometry - A Self Guided Study Companion

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+--------+-------------------------+-------------------------+--------+--------+
|00000000| 54 55 54 4f 52 20 30 36 | 7c 20 00 00 d6 00 00 00 |TUTOR 06|| ......|
|00000010| 53 65 63 74 69 6f 6e 20 | 32 2e 34 20 20 54 72 61 |Section |2.4  Tra|
|00000020| 6e 73 6c 61 74 69 6f 6e | 73 20 61 6e 64 20 43 6f |nslation|s and Co|
|00000030| 6d 62 69 6e 61 74 69 6f | 6e 73 0d 0a 00 0d 0a 00 |mbinatio|ns......|
|00000040| 0e 65 32 2d 34 2d 31 0e | 47 75 69 64 65 64 20 45 |.e2-4-1.|Guided E|
|00000050| 78 61 6d 70 6c 65 20 31 | 0f 20 20 46 69 6e 64 69 |xample 1|.  Findi|
|00000060| 6e 67 20 43 6f 6d 62 69 | 6e 61 74 69 6f 6e 73 20 |ng Combi|nations |
|00000070| 6f 66 20 46 75 6e 63 74 | 69 6f 6e 73 0d 0a 00 0d |of Funct|ions....|
|00000080| 0b 00 0e 65 32 2d 34 2d | 32 0e 47 75 69 64 65 64 |...e2-4-|2.Guided|
|00000090| 20 45 78 61 6d 70 6c 65 | 20 32 0f 20 20 45 76 61 | Example| 2.  Eva|
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|000000d0| 75 69 64 65 64 20 45 78 | 61 6d 70 6c 65 20 33 0f |uided Ex|ample 3.|
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|000002a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 33 0d |        |    .23.|
|000002b0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000002c0| 11 33 66 11 31 28 11 33 | 78 11 31 29 20 3d 20 11 |.3f.1(.3|x.1) = .|
|000002d0| 33 78 20 20 20 20 20 20 | 11 31 61 6e 64 20 20 20 |3x      |.1and   |
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|000002f0| 20 11 33 78 20 11 31 2b | 20 31 20 2e 0d 0a 00 0d | .3x .1+| 1 .....|
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|00000380| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000390| 20 20 11 32 33 0d 0b 00 | 20 20 20 20 20 20 20 20 |  .23...|        |
|000003a0| 20 20 20 20 20 20 20 20 | 20 20 11 31 3d 20 11 33 |        |  .1= .3|
|000003b0| 78 20 20 11 31 2b 20 11 | 33 78 20 11 31 2b 20 31 |x  .1+ .|3x .1+ 1|
|000003c0| 20 2e 13 0d 0a 00 0d 0b | 00 54 68 65 20 64 69 66 | .......|.The dif|
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|000003e0| 31 61 6e 64 20 11 33 67 | 20 11 31 69 73 20 67 69 |1and .3g| .1is gi|
|000003f0| 76 65 6e 20 62 79 0d 0a | 00 0d 0b 00 20 20 20 20 |ven by..|....    |
|00000400| 20 20 20 28 11 33 66 20 | 11 31 2d 20 11 33 67 11 |   (.3f |.1- .3g.|
|00000410| 31 29 28 11 33 78 11 31 | 29 20 3d 20 11 33 66 11 |1)(.3x.1|) = .3f.|
|00000420| 31 28 11 33 78 11 31 29 | 20 2d 20 11 33 67 11 31 |1(.3x.1)| - .3g.1|
|00000430| 28 11 33 78 11 31 29 13 | 0d 0a 00 20 20 20 20 20 |(.3x.1).|...     |
|00000440| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000450| 11 32 33 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |.23...  |        |
|00000460| 20 20 20 20 20 20 20 20 | 11 31 3d 20 11 33 78 20 |        |.1= .3x |
|00000470| 20 11 31 2d 20 28 11 33 | 78 20 11 31 2b 20 31 29 | .1- (.3|x .1+ 1)|
|00000480| 13 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|00000490| 20 20 20 20 20 20 20 20 | 20 11 32 33 0d 0b 00 20 |        | .23... |
|000004a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000004b0| 20 11 31 3d 20 11 33 78 | 20 20 11 31 2d 20 11 33 | .1= .3x|  .1- .3|
|000004c0| 78 20 11 31 2d 20 31 20 | 2e 13 0d 0a 00 0d 0b 00 |x .1- 1 |........|
|000004d0| 54 68 65 20 70 72 6f 64 | 75 63 74 20 6f 66 20 11 |The prod|uct of .|
|000004e0| 33 66 20 11 31 61 6e 64 | 20 11 33 67 20 11 31 69 |3f .1and| .3g .1i|
|000004f0| 73 20 67 69 76 65 6e 20 | 62 79 0d 0a 00 0d 0b 00 |s given |by......|
|00000500| 20 20 20 20 20 20 20 20 | 20 20 28 11 33 66 67 11 |        |  (.3fg.|
|00000510| 31 29 28 11 33 78 11 31 | 29 20 3d 20 11 33 66 11 |1)(.3x.1|) = .3f.|
|00000520| 31 28 11 33 78 11 31 29 | 11 34 2a 11 33 67 11 31 |1(.3x.1)|.4*.3g.1|
|00000530| 28 11 33 78 11 31 29 0d | 0a 00 20 20 20 20 20 20 |(.3x.1).|..      |
|00000540| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00000550| 32 33 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |23...   |        |
|00000560| 20 20 20 20 20 20 20 11 | 31 3d 20 11 33 78 20 11 |       .|1= .3x .|
|00000570| 31 28 11 33 78 20 11 31 | 2b 20 31 29 13 0d 0a 00 |1(.3x .1|+ 1)....|
|00000580| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000590| 20 20 20 20 20 11 32 34 | 20 20 20 20 33 0d 0b 00 |     .24|    3...|
|000005a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000005b0| 20 20 11 31 3d 20 11 33 | 78 20 20 11 31 2b 20 11 |  .1= .3|x  .1+ .|
|000005c0| 33 78 20 20 11 31 2e 13 | 0d 0a 00 0d 0b 00 54 68 |3x  .1..|......Th|
|000005d0| 65 20 71 75 6f 74 69 65 | 6e 74 20 6f 66 20 11 33 |e quotie|nt of .3|
|000005e0| 66 20 11 31 61 6e 64 20 | 11 33 67 20 11 31 69 73 |f .1and |.3g .1is|
|000005f0| 20 67 69 76 65 6e 20 62 | 79 0d 0a 00 20 20 20 20 | given b|y...    |
|00000600| 20 20 20 20 20 20 20 11 | 34 28 20 29 0d 0b 00 20 |       .|4( )... |
|00000610| 20 20 20 20 20 20 20 20 | 20 20 21 11 33 66 11 34 |        |  !.3f.4|
|00000620| 21 20 20 20 20 20 20 11 | 33 66 11 31 28 11 33 78 |!      .|3f.1(.3x|
|00000630| 11 31 29 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |.1)...  |        |
|00000640| 20 11 34 21 32 21 11 31 | 28 11 33 78 11 31 29 20 | .4!2!.1|(.3x.1) |
|00000650| 3d 20 11 34 32 32 32 32 | 0d 0b 00 20 20 20 20 20 |= .42222|...     |
|00000660| 20 20 20 20 20 20 21 11 | 33 67 11 34 21 20 20 20 |      !.|3g.4!   |
|00000670| 20 20 20 11 33 67 11 31 | 28 11 33 78 11 31 29 0d |   .3g.1|(.3x.1).|
|00000680| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 11 34 39 |..      |     .49|
|00000690| 20 30 20 20 20 20 20 20 | 20 20 20 20 11 31 13 0d | 0      |    .1..|
|000006a0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000006b0| 20 20 20 20 20 20 20 20 | 20 11 32 33 0d 0b 00 20 |        | .23... |
|000006c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000006d0| 20 20 20 20 20 11 33 78 | 0d 0b 00 20 20 20 20 20 |     .3x|...     |
|000006e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 3d |        |     .1=|
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|00000700| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000710| 20 20 20 11 33 78 20 11 | 31 2b 20 31 0d 0a 00 53 |   .3x .|1+ 1...S|
|00000720| 65 63 74 69 6f 6e 20 32 | 2e 34 20 20 54 72 61 6e |ection 2|.4  Tran|
|00000730| 73 6c 61 74 69 6f 6e 73 | 20 61 6e 64 20 43 6f 6d |slations| and Com|
|00000740| 62 69 6e 61 74 69 6f 6e | 73 0d 0b 00 20 20 20 20 |bination|s...    |
|00000750| 20 20 20 20 20 20 20 20 | 11 32 32 0d 0b 00 11 31 |        |.22....1|
|00000760| 4c 65 74 20 11 33 66 11 | 31 28 11 33 78 11 31 29 |Let .3f.|1(.3x.1)|
|00000770| 20 3d 20 11 33 78 20 20 | 11 31 2b 20 34 20 61 6e | = .3x  |.1+ 4 an|
|00000780| 64 20 6c 65 74 20 11 33 | 67 11 31 28 11 33 78 11 |d let .3|g.1(.3x.|
|00000790| 31 29 20 3d 20 11 33 78 | 20 11 31 2d 20 32 2e 20 |1) = .3x| .1- 2. |
|000007a0| 20 45 76 61 6c 75 61 74 | 65 20 74 68 65 20 66 6f | Evaluat|e the fo|
|000007b0| 6c 6c 6f 77 69 6e 67 2e | 0d 0a 00 0d 0b 00 28 61 |llowing.|......(a|
|000007c0| 29 20 20 28 11 33 66 20 | 11 31 2d 20 11 33 67 11 |)  (.3f |.1- .3g.|
|000007d0| 31 29 28 32 11 33 78 11 | 31 29 20 20 20 20 20 20 |1)(2.3x.|1)      |
|000007e0| 20 20 20 20 20 20 20 20 | 20 20 20 28 62 29 20 20 |        |   (b)  |
|000007f0| 28 11 33 66 67 11 31 29 | 28 33 29 20 2d 20 11 33 |(.3fg.1)|(3) - .3|
|00000800| 66 11 31 28 2d 31 29 0d | 0a 00 0d 0b 00 13 12 31 |f.1(-1).|.......1|
|00000810| 53 4f 4c 55 54 49 4f 4e | 12 30 0d 0a 00 0d 0b 00 |SOLUTION|.0......|
|00000820| 61 29 20 54 68 65 20 64 | 69 66 66 65 72 65 6e 63 |a) The d|ifferenc|
|00000830| 65 20 6f 66 20 74 68 65 | 20 66 75 6e 63 74 69 6f |e of the| functio|
|00000840| 6e 73 20 11 33 66 20 11 | 31 61 6e 64 20 11 33 67 |ns .3f .|1and .3g|
|00000850| 20 11 31 69 73 20 67 69 | 76 65 6e 20 62 79 0d 0a | .1is gi|ven by..|
|00000860| 00 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|00000870| 20 20 20 20 28 11 33 66 | 20 11 31 2d 20 11 33 67 |    (.3f| .1- .3g|
|00000880| 11 31 29 28 11 33 78 11 | 31 29 20 3d 20 11 33 66 |.1)(.3x.|1) = .3f|
|00000890| 11 31 28 11 33 78 11 31 | 29 20 2d 20 11 33 67 11 |.1(.3x.1|) - .3g.|
|000008a0| 31 28 11 33 78 11 31 29 | 13 0d 0a 00 20 20 20 20 |1(.3x.1)|....    |
|000008b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000008c0| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|000008d0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|000008e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 3d 20 |        |    .1= |
|000008f0| 28 11 33 78 20 20 11 31 | 2b 20 34 29 20 2d 20 28 |(.3x  .1|+ 4) - (|
|00000900| 11 33 78 20 11 31 2d 20 | 32 29 13 0d 0a 00 20 20 |.3x .1- |2)....  |
|00000910| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000920| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 32 0d |        |    .22.|
|00000930| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00000940| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 3d |        |     .1=|
|00000950| 20 11 33 78 20 20 11 31 | 2d 20 11 33 78 20 11 31 | .3x  .1|- .3x .1|
|00000960| 2b 20 36 2e 13 0d 0a 00 | 0d 0b 00 20 20 20 54 68 |+ 6.....|...   Th|
|00000970| 65 72 65 66 6f 72 65 2c | 20 28 11 33 66 20 11 31 |erefore,| (.3f .1|
|00000980| 2d 20 11 33 67 11 31 29 | 28 32 11 33 78 11 31 29 |- .3g.1)|(2.3x.1)|
|00000990| 20 69 73 20 67 69 76 65 | 6e 20 62 79 0d 0a 00 20 | is give|n by... |
|000009a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000009b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000009c0| 11 32 32 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |.22...  |        |
|000009d0| 20 20 20 20 20 11 31 28 | 11 33 66 20 11 31 2d 20 |     .1(|.3f .1- |
|000009e0| 11 33 67 11 31 29 28 32 | 11 33 78 11 31 29 20 3d |.3g.1)(2|.3x.1) =|
|000009f0| 20 28 32 11 33 78 11 31 | 29 20 20 2d 20 32 11 33 | (2.3x.1|)  - 2.3|
|00000a00| 78 20 11 31 2b 20 36 13 | 0d 0a 00 20 20 20 20 20 |x .1+ 6.|...     |
|00000a10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000a20| 20 20 20 20 20 20 20 20 | 20 20 11 32 32 0d 0b 00 |        |  .22...|
|00000a30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000a40| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 3d 20 34 |        |   .1= 4|
|00000a50| 11 33 78 20 20 11 31 2d | 20 32 11 33 78 20 11 31 |.3x  .1-| 2.3x .1|
|00000a60| 2b 20 36 2e 13 0d 0a 00 | 0d 0b 00 62 29 20 54 68 |+ 6.....|...b) Th|
|00000a70| 65 20 70 72 6f 64 75 63 | 74 20 6f 66 20 74 68 65 |e produc|t of the|
|00000a80| 20 66 75 6e 63 74 69 6f | 6e 73 20 11 33 66 20 11 | functio|ns .3f .|
|00000a90| 31 61 6e 64 20 11 33 67 | 20 11 31 69 73 20 67 69 |1and .3g| .1is gi|
|00000aa0| 76 65 6e 20 62 79 0d 0a | 00 0d 0b 00 20 20 20 20 |ven by..|....    |
|00000ab0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 28 |        |       (|
|00000ac0| 11 33 66 67 11 31 29 28 | 11 33 78 11 31 29 20 3d |.3fg.1)(|.3x.1) =|
|00000ad0| 20 11 33 66 11 31 28 11 | 33 78 11 31 29 11 34 2a | .3f.1(.|3x.1).4*|
|00000ae0| 11 33 67 11 31 28 11 33 | 78 11 31 29 0d 0a 00 20 |.3g.1(.3|x.1)... |
|00000af0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000b00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00000b10| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |2...    |        |
|00000b20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00000b30| 31 3d 20 28 11 33 78 20 | 20 11 31 2b 20 34 29 28 |1= (.3x | .1+ 4)(|
|00000b40| 11 33 78 20 11 31 2d 20 | 32 29 13 0d 0a 00 20 20 |.3x .1- |2)....  |
|00000b50| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000b60| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 33 20 |        |    .23 |
|00000b70| 20 20 20 20 32 0d 0b 00 | 20 20 20 20 20 20 20 20 |    2...|        |
|00000b80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000b90| 20 20 20 11 31 3d 20 11 | 33 78 20 20 11 31 2d 20 |   .1= .|3x  .1- |
|00000ba0| 32 11 33 78 20 20 11 31 | 2b 20 34 11 33 78 20 11 |2.3x  .1|+ 4.3x .|
|00000bb0| 31 2d 20 38 2e 13 0d 0a | 00 0d 0b 00 20 20 20 54 |1- 8....|....   T|
|00000bc0| 68 65 72 65 66 6f 72 65 | 2c 20 28 11 33 66 67 11 |herefore|, (.3fg.|
|00000bd0| 31 29 28 33 29 20 69 73 | 20 67 69 76 65 6e 20 62 |1)(3) is| given b|
|00000be0| 79 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |y...    |        |
|00000bf0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000c00| 20 20 11 32 33 20 20 20 | 20 20 20 20 32 0d 0b 00 |  .23   |    2...|
|00000c10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000c20| 20 20 20 11 31 28 11 33 | 66 67 11 31 29 28 33 29 |   .1(.3|fg.1)(3)|
|00000c30| 20 3d 20 33 20 20 2d 20 | 32 28 33 29 20 20 2b 20 | = 3  - |2(3)  + |
|00000c40| 34 28 33 29 20 2d 20 38 | 13 0d 0a 00 0d 0b 00 20 |4(3) - 8|....... |
|00000c50| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000c60| 20 20 20 20 20 20 20 20 | 20 20 3d 20 32 37 20 2d |        |  = 27 -|
|00000c70| 20 31 38 20 2b 20 31 32 | 20 2d 20 38 13 0d 0a 00 | 18 + 12| - 8....|
|00000c80| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00000c90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 3d 20 |        |      = |
|00000ca0| 31 33 2e 13 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |13..... |        |
|00000cb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 32 0d |        |    .22.|
|00000cc0| 0b 00 20 20 20 11 31 53 | 69 6e 63 65 20 11 33 66 |..   .1S|ince .3f|
|00000cd0| 11 31 28 2d 31 29 20 3d | 20 28 2d 31 29 20 20 2b |.1(-1) =| (-1)  +|
|00000ce0| 20 34 20 3d 20 35 2c 20 | 77 65 20 68 61 76 65 0d | 4 = 5, |we have.|
|00000cf0| 0a 00 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |.....   |        |
|00000d00| 28 11 33 66 67 11 31 29 | 28 33 29 20 2d 20 11 33 |(.3fg.1)|(3) - .3|
|00000d10| 66 11 31 28 2d 31 29 20 | 3d 20 31 33 20 2d 20 35 |f.1(-1) |= 13 - 5|
|00000d20| 13 0d 0a 00 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |....... |        |
|00000d30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000d40| 20 20 3d 20 38 2e 0d 0a | 00 53 65 63 74 69 6f 6e |  = 8...|.Section|
|00000d50| 20 32 2e 34 20 20 54 72 | 61 6e 73 6c 61 74 69 6f | 2.4  Tr|anslatio|
|00000d60| 6e 73 20 61 6e 64 20 43 | 6f 6d 62 69 6e 61 74 69 |ns and C|ombinati|
|00000d70| 6f 6e 73 0d 0b 00 4c 65 | 74 20 11 33 66 11 31 28 |ons...Le|t .3f.1(|
|00000d80| 11 33 78 11 31 29 20 3d | 20 11 33 78 20 11 31 2d |.3x.1) =| .3x .1-|
|00000d90| 20 32 20 61 6e 64 20 11 | 33 67 11 31 28 11 33 78 | 2 and .|3g.1(.3x|
|00000da0| 11 31 29 20 3d 20 32 11 | 33 78 20 11 31 2b 20 31 |.1) = 2.|3x .1+ 1|
|00000db0| 2e 20 20 46 69 6e 64 20 | 74 68 65 20 66 6f 6c 6c |.  Find |the foll|
|00000dc0| 6f 77 69 6e 67 2e 0d 0a | 00 0d 0b 00 28 61 29 20 |owing...|....(a) |
|00000dd0| 20 11 33 66 20 11 34 6f | 20 11 33 67 20 20 20 20 | .3f .4o| .3g    |
|00000de0| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 28 62 29 |        |   .1(b)|
|00000df0| 20 20 11 33 67 20 11 34 | 6f 20 11 33 66 20 20 20 |  .3g .4|o .3f   |
|00000e00| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 28 63 |        |    .1(c|
|00000e10| 29 20 20 11 33 66 20 11 | 34 6f 20 11 33 66 0d 0a |)  .3f .|4o .3f..|
|00000e20| 00 0d 0b 00 11 31 13 12 | 31 53 4f 4c 55 54 49 4f |.....1..|1SOLUTIO|
|00000e30| 4e 12 30 0d 0a 00 61 29 | 20 54 68 65 20 63 6f 6d |N.0...a)| The com|
|00000e40| 70 6f 73 69 74 69 6f 6e | 20 6f 66 20 11 33 66 20 |position| of .3f |
|00000e50| 11 31 77 69 74 68 20 11 | 33 67 20 11 31 69 73 20 |.1with .|3g .1is |
|00000e60| 61 73 20 66 6f 6c 6c 6f | 77 73 2e 0d 0a 00 0d 0b |as follo|ws......|
|00000e70| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 28 11 33 |.       |     (.3|
|00000e80| 66 20 11 34 6f 20 11 33 | 67 11 31 29 28 11 33 78 |f .4o .3|g.1)(.3x|
|00000e90| 11 31 29 20 3d 20 11 33 | 66 11 31 28 11 33 67 11 |.1) = .3|f.1(.3g.|
|00000ea0| 31 28 11 33 78 11 31 29 | 29 20 20 20 20 20 20 20 |1(.3x.1)|)       |
|00000eb0| 20 20 20 20 20 20 12 31 | 11 32 44 65 66 69 6e 69 |      .1|.2Defini|
|00000ec0| 74 69 6f 6e 20 6f 66 20 | 66 11 34 6f 11 32 67 11 |tion of |f.4o.2g.|
|00000ed0| 31 12 30 13 0d 0a 00 0d | 0b 00 20 20 20 20 20 20 |1.0.....|..      |
|00000ee0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000ef0| 20 3d 20 11 33 66 11 31 | 28 32 11 33 78 20 11 31 | = .3f.1|(2.3x .1|
|00000f00| 2b 20 31 29 20 20 20 20 | 20 20 20 20 20 20 20 12 |+ 1)    |       .|
|00000f10| 31 11 32 44 65 66 69 6e | 69 74 69 6f 6e 20 6f 66 |1.2Defin|ition of|
|00000f20| 20 67 28 78 29 11 31 12 | 30 13 0d 0a 00 0d 0b 00 | g(x).1.|0.......|
|00000f30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000f40| 20 20 20 20 20 20 20 3d | 20 28 32 11 33 78 20 11 |       =| (2.3x .|
|00000f50| 31 2b 20 31 29 20 2d 20 | 32 20 20 20 20 20 20 20 |1+ 1) - |2       |
|00000f60| 20 12 31 11 32 44 65 66 | 69 6e 69 74 69 6f 6e 20 | .1.2Def|inition |
|00000f70| 6f 66 20 66 28 78 29 11 | 31 12 30 13 0d 0a 00 0d |of f(x).|1.0.....|
|00000f80| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00000f90| 20 20 20 20 20 20 20 20 | 20 3d 20 32 11 33 78 20 |        | = 2.3x |
|00000fa0| 11 31 2d 20 31 13 0d 0a | 00 0d 0b 00 62 29 20 54 |.1- 1...|....b) T|
|00000fb0| 68 65 20 63 6f 6d 70 6f | 73 69 74 69 6f 6e 20 6f |he compo|sition o|
|00000fc0| 66 20 11 33 67 20 11 31 | 77 69 74 68 20 11 33 66 |f .3g .1|with .3f|
|00000fd0| 20 11 31 69 73 20 61 73 | 20 66 6f 6c 6c 6f 77 73 | .1is as| follows|
|00000fe0| 2e 0d 0a 00 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |....... |        |
|00000ff0| 20 20 20 28 11 33 67 20 | 11 34 6f 20 11 33 66 11 |   (.3g |.4o .3f.|
|00001000| 31 29 28 11 33 78 11 31 | 29 20 3d 20 11 33 67 11 |1)(.3x.1|) = .3g.|
|00001010| 31 28 11 33 66 11 31 28 | 11 33 78 11 31 29 29 20 |1(.3f.1(|.3x.1)) |
|00001020| 20 20 20 20 20 20 20 20 | 20 20 20 20 12 31 11 32 |        |    .1.2|
|00001030| 44 65 66 69 6e 69 74 69 | 6f 6e 20 6f 66 20 67 11 |Definiti|on of g.|
|00001040| 34 6f 11 32 66 11 31 12 | 30 13 0d 0a 00 0d 0b 00 |4o.2f.1.|0.......|
|00001050| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001060| 20 20 20 20 20 20 20 3d | 20 11 33 67 11 31 28 11 |       =| .3g.1(.|
|00001070| 33 78 20 11 31 2d 20 32 | 29 20 20 20 20 20 20 20 |3x .1- 2|)       |
|00001080| 20 20 20 20 20 12 31 11 | 32 44 65 66 69 6e 69 74 |     .1.|2Definit|
|00001090| 69 6f 6e 20 6f 66 20 66 | 28 78 29 11 31 12 30 13 |ion of f|(x).1.0.|
|000010a0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|000010b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 3d 20 32 |        |     = 2|
|000010c0| 28 11 33 78 20 11 31 2d | 20 32 29 20 2b 20 31 20 |(.3x .1-| 2) + 1 |
|000010d0| 20 20 20 20 20 20 20 12 | 31 11 32 44 65 66 69 6e |       .|1.2Defin|
|000010e0| 69 74 69 6f 6e 20 6f 66 | 20 67 28 78 29 11 31 12 |ition of| g(x).1.|
|000010f0| 30 0d 0a 00 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |0...... |        |
|00001100| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 3d 20 |        |      = |
|00001110| 32 11 33 78 20 11 31 2d | 20 34 20 2b 20 31 13 0d |2.3x .1-| 4 + 1..|
|00001120| 0a 00 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |.....   |        |
|00001130| 20 20 20 20 20 20 20 20 | 20 20 20 20 3d 20 32 11 |        |    = 2.|
|00001140| 33 78 20 11 31 2d 20 33 | 13 0d 0a 00 0d 0b 00 63 |3x .1- 3|.......c|
|00001150| 29 20 54 68 65 20 63 6f | 6d 70 6f 73 69 74 69 6f |) The co|mpositio|
|00001160| 6e 20 6f 66 20 11 33 66 | 20 11 31 77 69 74 68 20 |n of .3f| .1with |
|00001170| 11 33 66 20 11 31 69 73 | 20 61 73 20 66 6f 6c 6c |.3f .1is| as foll|
|00001180| 6f 77 73 2e 0d 0a 00 0d | 0b 00 20 20 20 20 20 20 |ows.....|..      |
|00001190| 20 20 20 20 20 20 28 11 | 33 66 20 11 34 6f 20 11 |      (.|3f .4o .|
|000011a0| 33 66 11 31 29 28 11 33 | 78 11 31 29 20 3d 20 11 |3f.1)(.3|x.1) = .|
|000011b0| 33 66 11 31 28 11 33 66 | 11 31 28 11 33 78 11 31 |3f.1(.3f|.1(.3x.1|
|000011c0| 29 29 20 20 20 20 20 20 | 20 20 20 20 20 20 20 12 |))      |       .|
|000011d0| 31 11 32 44 65 66 69 6e | 69 74 69 6f 6e 20 6f 66 |1.2Defin|ition of|
|000011e0| 20 66 11 34 6f 11 32 66 | 11 31 12 30 13 0d 0a 00 | f.4o.2f|.1.0....|
|000011f0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00001200| 20 20 20 20 20 20 20 20 | 20 20 3d 20 11 33 66 11 |        |  = .3f.|
|00001210| 31 28 11 33 78 20 11 31 | 2d 20 32 29 20 20 20 20 |1(.3x .1|- 2)    |
|00001220| 20 20 20 20 20 20 20 20 | 12 31 11 32 44 65 66 69 |        |.1.2Defi|
|00001230| 6e 69 74 69 6f 6e 20 6f | 66 20 66 28 78 29 11 31 |nition o|f f(x).1|
|00001240| 12 30 13 0d 0a 00 0d 0b | 00 20 20 20 20 20 20 20 |.0......|.       |
|00001250| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001260| 3d 20 28 11 33 78 20 11 | 31 2d 20 32 29 20 2d 20 |= (.3x .|1- 2) - |
|00001270| 32 20 20 20 20 20 20 20 | 20 20 12 31 11 32 44 65 |2       |  .1.2De|
|00001280| 66 69 6e 69 74 69 6f 6e | 20 6f 66 20 66 28 78 29 |finition| of f(x)|
|00001290| 11 31 12 30 13 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |.1.0....|...     |
|000012a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000012b0| 20 20 3d 20 11 33 78 20 | 11 31 2d 20 34 0d 0a 00 |  = .3x |.1- 4...|
|000012c0| 53 65 63 74 69 6f 6e 20 | 32 2e 34 20 20 54 72 61 |Section |2.4  Tra|
|000012d0| 6e 73 6c 61 74 69 6f 6e | 73 20 61 6e 64 20 43 6f |nslation|s and Co|
|000012e0| 6d 62 69 6e 61 74 69 6f | 6e 73 0d 0b 00 20 20 20 |mbinatio|ns...   |
|000012f0| 20 20 20 20 20 20 20 20 | 20 11 34 44 32 32 32 32 |        | .4D2222|
|00001300| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |2       |      .2|
|00001310| 32 0d 0b 00 11 31 4c 65 | 74 20 11 33 66 11 31 28 |2....1Le|t .3f.1(|
|00001320| 11 33 78 11 31 29 20 3d | 20 11 34 53 20 11 33 78 |.3x.1) =| .4S .3x|
|00001330| 20 11 31 2b 20 32 20 61 | 6e 64 20 11 33 67 11 31 | .1+ 2 a|nd .3g.1|
|00001340| 28 11 33 78 11 31 29 20 | 3d 20 11 33 78 20 11 31 |(.3x.1) |= .3x .1|
|00001350| 2e 20 20 46 69 6e 64 20 | 11 33 66 20 11 34 6f 20 |.  Find |.3f .4o |
|00001360| 11 33 67 20 11 31 61 6e | 64 20 11 33 67 20 11 34 |.3g .1an|d .3g .4|
|00001370| 6f 20 11 33 66 11 31 2e | 0d 0a 00 0d 0b 00 13 12 |o .3f.1.|........|
|00001380| 31 53 4f 4c 55 54 49 4f | 4e 12 30 0d 0a 00 54 68 |1SOLUTIO|N.0...Th|
|00001390| 65 20 63 6f 6d 70 6f 73 | 69 74 69 6f 6e 20 6f 66 |e compos|ition of|
|000013a0| 20 11 33 66 20 11 31 77 | 69 74 68 20 11 33 67 20 | .3f .1w|ith .3g |
|000013b0| 11 31 69 73 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |.1is as |follows.|
|000013c0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|000013d0| 20 20 28 11 33 66 20 11 | 34 6f 20 11 33 67 11 31 |  (.3f .|4o .3g.1|
|000013e0| 29 28 11 33 78 11 31 29 | 20 3d 20 11 33 66 11 31 |)(.3x.1)| = .3f.1|
|000013f0| 28 11 33 67 11 31 28 11 | 33 78 11 31 29 29 20 20 |(.3g.1(.|3x.1))  |
|00001400| 20 20 20 20 20 20 20 20 | 20 20 20 12 31 11 32 44 |        |   .1.2D|
|00001410| 65 66 69 6e 69 74 69 6f | 6e 20 6f 66 20 66 11 34 |efinitio|n of f.4|
|00001420| 6f 11 32 67 11 31 12 30 | 13 0d 0a 00 0d 0b 00 20 |o.2g.1.0|....... |
|00001430| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001440| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|00001450| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001460| 20 20 20 20 20 20 20 20 | 11 31 3d 20 11 33 66 11 |        |.1= .3f.|
|00001470| 31 28 11 33 78 20 11 31 | 29 20 20 20 20 20 20 20 |1(.3x .1|)       |
|00001480| 20 20 20 20 20 20 20 20 | 12 31 11 32 44 65 66 69 |        |.1.2Defi|
|00001490| 6e 69 74 69 6f 6e 20 6f | 66 20 67 28 78 29 11 31 |nition o|f g(x).1|
|000014a0| 12 30 13 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |.0....  |        |
|000014b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000014c0| 20 11 34 67 32 32 32 32 | 32 0d 0b 00 20 20 20 20 | .4g2222|2...    |
|000014d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000014e0| 20 20 20 20 20 20 66 20 | 11 32 32 0d 0b 00 20 20 |      f |.22...  |
|000014f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001500| 20 20 20 20 20 11 31 3d | 20 11 34 76 20 11 33 78 |     .1=| .4v .3x|
|00001510| 20 20 11 31 2b 20 32 20 | 20 20 20 20 20 20 20 20 |  .1+ 2 |        |
|00001520| 20 20 20 12 31 11 32 44 | 65 66 69 6e 69 74 69 6f |   .1.2D|efinitio|
|00001530| 6e 20 6f 66 20 66 28 78 | 29 11 31 12 30 13 0d 0a |n of f(x|).1.0...|
|00001540| 00 0d 0b 00 54 68 65 20 | 63 6f 6d 70 6f 73 69 74 |....The |composit|
|00001550| 69 6f 6e 20 6f 66 20 11 | 33 67 20 11 31 77 69 74 |ion of .|3g .1wit|
|00001560| 68 20 11 33 66 20 11 31 | 69 73 20 61 73 20 66 6f |h .3f .1|is as fo|
|00001570| 6c 6c 6f 77 73 2e 0d 0a | 00 0d 0b 00 20 20 20 20 |llows...|....    |
|00001580| 20 20 20 20 20 20 20 20 | 28 11 33 67 20 11 34 6f |        |(.3g .4o|
|00001590| 20 11 33 66 11 31 29 28 | 11 33 78 11 31 29 20 3d | .3f.1)(|.3x.1) =|
|000015a0| 20 11 33 67 11 31 28 11 | 33 66 11 31 28 11 33 78 | .3g.1(.|3f.1(.3x|
|000015b0| 11 31 29 29 20 20 20 20 | 20 20 20 20 20 20 20 20 |.1))    |        |
|000015c0| 20 12 31 11 32 44 65 66 | 69 6e 69 74 69 6f 6e 20 | .1.2Def|inition |
|000015d0| 6f 66 20 67 11 34 6f 11 | 32 66 11 31 12 30 13 0d |of g.4o.|2f.1.0..|
|000015e0| 0a 00 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |.....   |        |
|000015f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001600| 20 11 34 44 32 32 32 32 | 32 0d 0b 00 20 20 20 20 | .4D2222|2...    |
|00001610| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001620| 20 20 20 11 31 3d 20 11 | 33 67 11 31 28 11 34 53 |   .1= .|3g.1(.4S|
|00001630| 20 11 33 78 20 11 31 2b | 20 32 29 20 20 20 20 20 | .3x .1+| 2)     |
|00001640| 20 20 20 20 20 12 31 11 | 32 44 65 66 69 6e 69 74 |     .1.|2Definit|
|00001650| 69 6f 6e 20 6f 66 20 66 | 28 78 29 11 31 12 30 13 |ion of f|(x).1.0.|
|00001660| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|00001670| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001680| 20 11 34 44 32 32 32 32 | 32 20 11 32 32 0d 0b 00 | .4D2222|2 .22...|
|00001690| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000016a0| 20 20 20 20 20 20 20 11 | 31 3d 20 28 11 34 53 20 |       .|1= (.4S |
|000016b0| 11 33 78 20 11 31 2b 20 | 32 29 20 20 20 20 20 20 |.3x .1+ |2)      |
|000016c0| 20 20 20 20 20 12 31 11 | 32 44 65 66 69 6e 69 74 |     .1.|2Definit|
|000016d0| 69 6f 6e 20 6f 66 20 67 | 28 78 29 11 31 12 30 13 |ion of g|(x).1.0.|
|000016e0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |......  |        |
|000016f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 3d 20 11 |        |     = .|
|00001700| 33 78 20 11 31 2b 20 32 | 20 2c 20 20 11 33 78 20 |3x .1+ 2| ,  .3x |
|00001710| 11 34 3e 20 11 31 2d 32 | 0d 0a 00 53 65 63 74 69 |.4> .1-2|...Secti|
|00001720| 6f 6e 20 32 2e 34 20 20 | 54 72 61 6e 73 6c 61 74 |on 2.4  |Translat|
|00001730| 69 6f 6e 73 20 61 6e 64 | 20 43 6f 6d 62 69 6e 61 |ions and| Combina|
|00001740| 74 69 6f 6e 73 0d 0b 00 | 4c 65 74 20 11 33 66 11 |tions...|Let .3f.|
|00001750| 31 28 11 33 78 11 31 29 | 20 3d 20 2d 32 7c 11 33 |1(.3x.1)| = -2|.3|
|00001760| 78 11 31 7c 20 61 6e 64 | 20 11 33 67 11 31 28 11 |x.1| and| .3g.1(.|
|00001770| 33 78 11 31 29 20 3d 20 | 11 33 78 20 11 31 2b 20 |3x.1) = |.3x .1+ |
|00001780| 33 2e 20 20 46 69 6e 64 | 20 11 33 66 20 11 34 6f |3.  Find| .3f .4o|
|00001790| 20 11 33 67 20 11 31 61 | 6e 64 20 11 33 67 20 11 | .3g .1a|nd .3g .|
|000017a0| 34 6f 20 11 33 66 11 31 | 2e 0d 0a 00 0d 0b 00 13 |4o .3f.1|........|
|000017b0| 12 31 53 4f 4c 55 54 49 | 4f 4e 12 30 0d 0a 00 54 |.1SOLUTI|ON.0...T|
|000017c0| 68 65 20 63 6f 6d 70 6f | 73 69 74 69 6f 6e 20 6f |he compo|sition o|
|000017d0| 66 20 11 33 66 20 11 31 | 77 69 74 68 20 11 33 67 |f .3f .1|with .3g|
|000017e0| 20 11 31 69 73 20 61 73 | 20 66 6f 6c 6c 6f 77 73 | .1is as| follows|
|000017f0| 2e 0d 0a 00 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |....... |        |
|00001800| 20 20 20 28 11 33 66 20 | 11 34 6f 20 11 33 67 11 |   (.3f |.4o .3g.|
|00001810| 31 29 28 11 33 78 11 31 | 29 20 3d 20 11 33 66 11 |1)(.3x.1|) = .3f.|
|00001820| 31 28 11 33 67 11 31 28 | 11 33 78 11 31 29 29 20 |1(.3g.1(|.3x.1)) |
|00001830| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 12 31 11 |        |     .1.|
|00001840| 32 44 65 66 69 6e 69 74 | 69 6f 6e 20 6f 66 20 66 |2Definit|ion of f|
|00001850| 11 34 6f 11 32 67 11 31 | 12 30 13 0d 0a 00 0d 0a |.4o.2g.1|.0......|
|00001860| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001870| 20 20 20 20 20 20 20 20 | 3d 20 11 33 66 11 31 28 |        |= .3f.1(|
|00001880| 11 33 78 20 11 31 2b 20 | 33 29 20 20 20 20 20 20 |.3x .1+ |3)      |
|00001890| 20 20 20 20 20 20 20 12 | 31 11 32 44 65 66 69 6e |       .|1.2Defin|
|000018a0| 69 74 69 6f 6e 20 6f 66 | 20 67 28 78 29 11 31 12 |ition of| g(x).1.|
|000018b0| 30 13 0d 0a 00 0d 0a 00 | 20 20 20 20 20 20 20 20 |0.......|        |
|000018c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 3d |        |       =|
|000018d0| 20 2d 32 7c 11 33 78 20 | 11 31 2b 20 33 7c 20 20 | -2|.3x |.1+ 3|  |
|000018e0| 20 20 20 20 20 20 20 20 | 20 20 12 31 11 32 44 65 |        |  .1.2De|
|000018f0| 66 69 6e 69 74 69 6f 6e | 20 6f 66 20 66 28 78 29 |finition| of f(x)|
|00001900| 11 31 12 30 13 0d 0a 00 | 0d 0b 00 54 68 65 20 63 |.1.0....|...The c|
|00001910| 6f 6d 70 6f 73 69 74 69 | 6f 6e 20 6f 66 20 11 33 |ompositi|on of .3|
|00001920| 67 20 11 31 77 69 74 68 | 20 11 33 66 20 11 31 69 |g .1with| .3f .1i|
|00001930| 73 20 61 73 20 66 6f 6c | 6c 6f 77 73 2e 0d 0a 00 |s as fol|lows....|
|00001940| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 28 |...     |       (|
|00001950| 11 33 67 20 11 34 6f 20 | 11 33 66 11 31 29 28 11 |.3g .4o |.3f.1)(.|
|00001960| 33 78 11 31 29 20 3d 20 | 11 33 67 11 31 28 11 33 |3x.1) = |.3g.1(.3|
|00001970| 66 11 31 28 11 33 78 11 | 31 29 29 20 20 20 20 20 |f.1(.3x.|1))     |
|00001980| 20 20 20 20 20 20 20 20 | 20 12 31 11 32 44 65 66 |        | .1.2Def|
|00001990| 69 6e 69 74 69 6f 6e 20 | 6f 66 20 67 11 34 6f 11 |inition |of g.4o.|
|000019a0| 32 66 11 31 12 30 13 0d | 0a 00 0d 0a 00 20 20 20 |2f.1.0..|.....   |
|000019b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000019c0| 20 20 20 20 3d 20 11 33 | 67 11 31 28 2d 32 7c 11 |    = .3|g.1(-2|.|
|000019d0| 33 78 11 31 7c 29 20 20 | 20 20 20 20 20 20 20 20 |3x.1|)  |        |
|000019e0| 20 20 20 12 31 11 32 44 | 65 66 69 6e 69 74 69 6f |   .1.2D|efinitio|
|000019f0| 6e 20 6f 66 20 66 28 78 | 29 11 31 12 30 13 0d 0a |n of f(x|).1.0...|
|00001a00| 00 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|00001a10| 20 20 20 20 20 20 20 20 | 20 20 20 3d 20 2d 32 7c |        |   = -2||
|00001a20| 11 33 78 11 31 7c 20 2b | 20 33 20 20 20 20 20 20 |.3x.1| +| 3      |
|00001a30| 20 20 20 20 20 20 12 31 | 11 32 44 65 66 69 6e 69 |      .1|.2Defini|
|00001a40| 74 69 6f 6e 20 6f 66 20 | 67 28 78 29 11 31 12 30 |tion of |g(x).1.0|
|00001a50| 0d 0a 00 53 65 63 74 69 | 6f 6e 20 32 2e 34 20 20 |...Secti|on 2.4  |
|00001a60| 54 72 61 6e 73 6c 61 74 | 69 6f 6e 73 20 61 6e 64 |Translat|ions and|
|00001a70| 20 43 6f 6d 62 69 6e 61 | 74 69 6f 6e 73 0d 0b 00 | Combina|tions...|
|00001a80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001a90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001aa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ab0| 20 20 20 20 20 20 20 20 | 20 11 32 32 0d 0b 00 11 |        | .22....|
|00001ac0| 31 46 69 6e 64 20 74 77 | 6f 20 66 75 6e 63 74 69 |1Find tw|o functi|
|00001ad0| 6f 6e 73 20 11 33 66 20 | 11 31 61 6e 64 20 11 33 |ons .3f |.1and .3|
|00001ae0| 67 20 11 31 73 75 63 68 | 20 74 68 61 74 20 28 11 |g .1such| that (.|
|00001af0| 33 66 20 11 34 6f 20 11 | 33 67 11 31 29 28 11 33 |3f .4o .|3g.1)(.3|
|00001b00| 78 11 31 29 20 3d 20 28 | 11 33 78 20 11 31 2d 20 |x.1) = (|.3x .1- |
|00001b10| 32 29 20 2e 20 20 28 4e | 6f 74 65 3a 20 20 54 68 |2) .  (N|ote:  Th|
|00001b20| 65 72 65 0d 0a 00 61 72 | 65 20 6d 61 6e 79 20 63 |ere...ar|e many c|
|00001b30| 6f 72 72 65 63 74 20 61 | 6e 73 77 65 72 73 2e 29 |orrect a|nswers.)|
|00001b40| 0d 0a 00 0d 0b 00 13 12 | 31 53 4f 4c 55 54 49 4f |........|1SOLUTIO|
|00001b50| 4e 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |N.0...  |        |
|00001b60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001b70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00001b80| 32 0d 0b 00 11 31 4f 6e | 65 20 70 6f 73 73 69 62 |2....1On|e possib|
|00001b90| 6c 65 20 73 6f 6c 75 74 | 69 6f 6e 20 69 73 20 74 |le solut|ion is t|
|00001ba0| 6f 20 6c 65 74 20 11 33 | 66 11 31 28 11 33 78 11 |o let .3|f.1(.3x.|
|00001bb0| 31 29 20 3d 20 11 33 78 | 20 20 11 31 61 6e 64 20 |1) = .3x|  .1and |
|00001bc0| 11 33 67 11 31 28 11 33 | 78 11 31 29 20 3d 20 11 |.3g.1(.3|x.1) = .|
|00001bd0| 33 78 20 11 31 2d 20 32 | 2e 20 20 54 68 65 6e 20 |3x .1- 2|.  Then |
|00001be0| 74 68 65 0d 0a 00 63 6f | 6d 70 6f 73 69 74 69 6f |the...co|mpositio|
|00001bf0| 6e 20 6f 66 20 11 33 66 | 20 11 31 77 69 74 68 20 |n of .3f| .1with |
|00001c00| 11 33 67 20 11 31 69 73 | 20 67 69 76 65 6e 20 62 |.3g .1is| given b|
|00001c10| 79 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |y...    |        |
|00001c20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001c30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001c40| 20 20 11 32 32 0d 0b 00 | 20 20 20 20 20 11 31 28 |  .22...|     .1(|
|00001c50| 11 33 66 20 11 34 6f 20 | 11 33 67 11 31 29 28 11 |.3f .4o |.3g.1)(.|
|00001c60| 33 78 11 31 29 20 3d 20 | 11 33 66 11 31 28 11 33 |3x.1) = |.3f.1(.3|
|00001c70| 67 11 31 28 11 33 78 11 | 31 29 29 20 3d 20 11 33 |g.1(.3x.|1)) = .3|
|00001c80| 66 11 31 28 11 33 78 20 | 11 31 2d 20 32 29 20 3d |f.1(.3x |.1- 2) =|
|00001c90| 20 28 11 33 78 20 11 31 | 2d 20 32 29 20 2e 0d 0a | (.3x .1|- 2) ...|
|00001ca0| 00 53 65 63 74 69 6f 6e | 20 32 2e 34 20 20 54 72 |.Section| 2.4  Tr|
|00001cb0| 61 6e 73 6c 61 74 69 6f | 6e 73 20 61 6e 64 20 43 |anslatio|ns and C|
|00001cc0| 6f 6d 62 69 6e 61 74 69 | 6f 6e 73 0d 0b 00 46 69 |ombinati|ons...Fi|
|00001cd0| 6e 64 20 74 68 65 20 64 | 6f 6d 61 69 6e 20 6f 66 |nd the d|omain of|
|00001ce0| 20 11 33 66 11 31 2c 20 | 11 33 67 11 31 2c 20 61 | .3f.1, |.3g.1, a|
|00001cf0| 6e 64 20 11 33 66 20 11 | 34 6f 20 11 33 67 20 11 |nd .3f .|4o .3g .|
|00001d00| 31 67 69 76 65 6e 0d 0a | 00 20 20 20 20 20 20 20 |1given..|.       |
|00001d10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 34 |        |      .4|
|00001d20| 44 32 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |D2      |        |
|00001d30| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|00001d40| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |.       |      .3|
|00001d50| 66 11 31 28 11 33 78 11 | 31 29 20 3d 20 11 34 53 |f.1(.3x.|1) = .4S|
|00001d60| 20 11 33 78 20 20 20 20 | 20 20 20 11 31 61 6e 64 | .3x    |   .1and|
|00001d70| 20 20 20 20 20 20 20 11 | 33 67 11 31 28 11 33 78 |       .|3g.1(.3x|
|00001d80| 11 31 29 20 3d 20 11 33 | 78 20 11 31 2e 0d 0a 00 |.1) = .3|x .1....|
|00001d90| 0d 0b 00 13 12 31 53 4f | 4c 55 54 49 4f 4e 12 30 |.....1SO|LUTION.0|
|00001da0| 0d 0a 00 53 69 6e 63 65 | 20 77 65 20 63 61 6e 6e |...Since| we cann|
|00001db0| 6f 74 20 74 61 6b 65 20 | 74 68 65 20 73 71 75 61 |ot take |the squa|
|00001dc0| 72 65 20 72 6f 6f 74 20 | 6f 66 20 61 20 6e 65 67 |re root |of a neg|
|00001dd0| 61 74 69 76 65 20 6e 75 | 6d 62 65 72 2c 20 74 68 |ative nu|mber, th|
|00001de0| 65 20 64 6f 6d 61 69 6e | 20 6f 66 20 11 33 66 20 |e domain| of .3f |
|00001df0| 11 31 69 73 0d 0a 00 61 | 6c 6c 20 72 65 61 6c 20 |.1is...a|ll real |
|00001e00| 6e 75 6d 62 65 72 73 20 | 67 72 65 61 74 65 72 20 |numbers |greater |
|00001e10| 74 68 61 6e 20 6f 72 20 | 65 71 75 61 6c 20 74 6f |than or |equal to|
|00001e20| 20 7a 65 72 6f 2e 0d 0a | 00 0d 0b 00 20 20 20 20 | zero...|....    |
|00001e30| 20 20 20 20 20 20 20 20 | 20 20 20 20 44 6f 6d 61 |        |    Doma|
|00001e40| 69 6e 20 6f 66 20 11 33 | 66 11 31 3a 20 20 5b 30 |in of .3|f.1:  [0|
|00001e50| 2c 20 11 34 38 11 31 29 | 0d 0b 00 20 20 20 20 20 |, .48.1)|...     |
|00001e60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001e70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 13 |        |       .|
|00001e80| 0d 0a 00 54 68 65 20 64 | 6f 6d 61 69 6e 20 6f 66 |...The d|omain of|
|00001e90| 20 11 33 67 20 11 31 69 | 73 20 74 68 65 20 73 65 | .3g .1i|s the se|
|00001ea0| 74 20 6f 66 20 61 6c 6c | 20 72 65 61 6c 20 6e 75 |t of all| real nu|
|00001eb0| 6d 62 65 72 73 2e 0d 0a | 00 0d 0b 00 20 20 20 20 |mbers...|....    |
|00001ec0| 20 20 20 20 20 20 20 20 | 20 20 20 20 44 6f 6d 61 |        |    Doma|
|00001ed0| 69 6e 20 6f 66 20 11 33 | 67 11 31 3a 20 20 28 2d |in of .3|g.1:  (-|
|00001ee0| 11 34 38 11 31 2c 20 11 | 34 38 11 31 29 0d 0b 00 |.48.1, .|48.1)...|
|00001ef0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001f00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001f10| 20 20 20 20 20 13 0d 0a | 00 54 6f 20 64 65 74 65 |     ...|.To dete|
|00001f20| 72 6d 69 6e 65 20 74 68 | 65 20 64 6f 6d 61 69 6e |rmine th|e domain|
|00001f30| 20 6f 66 20 11 33 66 20 | 11 34 6f 20 11 33 67 11 | of .3f |.4o .3g.|
|00001f40| 31 2c 20 69 74 20 6d 61 | 79 20 62 65 20 68 65 6c |1, it ma|y be hel|
|00001f50| 70 66 75 6c 20 74 6f 20 | 66 69 72 73 74 20 63 61 |pful to |first ca|
|00001f60| 6c 63 75 6c 61 74 65 20 | 69 74 73 0d 0a 00 76 61 |lculate |its...va|
|00001f70| 6c 75 65 2e 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |lue.... |        |
|00001f80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001f90| 20 20 20 20 20 20 20 11 | 34 67 32 32 32 0d 0b 00 |       .|4g222...|
|00001fa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001fb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 66 |        |       f|
|00001fc0| 20 20 11 32 32 0d 0b 00 | 20 20 20 20 20 20 20 20 |  .22...|        |
|00001fd0| 20 20 20 20 11 33 66 20 | 11 34 6f 20 11 33 67 20 |    .3f |.4o .3g |
|00001fe0| 11 31 3d 20 11 33 66 11 | 31 28 11 33 67 11 31 28 |.1= .3f.|1(.3g.1(|
|00001ff0| 11 33 78 11 31 29 29 20 | 3d 20 11 34 76 20 11 31 |.3x.1)) |= .4v .1|
|00002000| 28 11 33 78 20 11 31 29 | 20 3d 20 7c 11 33 78 11 |(.3x .1)| = |.3x.|
|00002010| 31 7c 13 0d 0a 00 0d 0b | 00 46 72 6f 6d 20 6f 75 |1|......|.From ou|
|00002020| 72 20 63 61 6c 63 75 6c | 61 74 69 6f 6e 2c 20 77 |r calcul|ation, w|
|00002030| 65 20 73 65 65 20 74 68 | 61 74 20 74 68 65 20 64 |e see th|at the d|
|00002040| 6f 6d 61 69 6e 20 6f 66 | 20 11 33 66 20 11 34 6f |omain of| .3f .4o|
|00002050| 20 11 33 67 20 11 31 69 | 73 20 74 68 65 20 73 65 | .3g .1i|s the se|
|00002060| 74 20 6f 66 20 61 6c 6c | 20 72 65 61 6c 20 0d 0a |t of all| real ..|
|00002070| 00 6e 75 6d 62 65 72 73 | 2e 0d 0a 00 3a 00 00 00 |.numbers|....:...|
|00002080| f9 01 00 00 4d 2a 00 00 | 10 00 00 00 00 00 00 00 |....M*..|........|
|00002090| 65 32 2d 34 00 5d 02 00 | 00 c2 04 00 00 4d 2a 00 |e2-4.]..|.....M*.|
|000020a0| 00 33 02 00 00 00 00 00 | 00 65 32 2d 34 2d 31 00 |.3......|.e2-4-1.|
|000020b0| 49 07 00 00 00 06 00 00 | 4d 2a 00 00 1f 07 00 00 |I.......|M*......|
|000020c0| 00 00 00 00 65 32 2d 34 | 2d 32 00 73 0d 00 00 4d |....e2-4|-2.s...M|
|000020d0| 05 00 00 4d 2a 00 00 49 | 0d 00 00 00 00 00 00 65 |...M*..I|.......e|
|000020e0| 32 2d 34 2d 33 00 ea 12 | 00 00 31 04 00 00 4d 2a |2-4-3...|..1...M*|
|000020f0| 00 00 c0 12 00 00 00 00 | 00 00 65 32 2d 34 2d 34 |........|..e2-4-4|
|00002100| 00 45 17 00 00 0e 03 00 | 00 4d 2a 00 00 1b 17 00 |.E......|.M*.....|
|00002110| 00 00 00 00 00 65 32 2d | 34 2d 35 00 7d 1a 00 00 |.....e2-|4-5.}...|
|00002120| 24 02 00 00 4d 2a 00 00 | 53 1a 00 00 00 00 00 00 |$...M*..|S.......|
|00002130| 69 32 2d 34 2d 31 00 cb | 1c 00 00 b1 03 00 00 4d |i2-4-1..|.......M|
|00002140| 2a 00 00 a1 1c 00 00 00 | 00 00 00 69 32 2d 34 2d |*.......|...i2-4-|
|00002150| 32 00                   |                         |2.      |        |
+--------+-------------------------+-------------------------+--------+--------+