Chapter 1: Worksheet 2 Jack K. Cohen Colorado School of Mines




Functions, Symbol Problems


Suggested Problems Section 1.1: 27, 45, 55

  1. It is important to understand the difference between a formula and a function. Some functions are specified by formulas—but there are also other ways to specify a function. And some formulas do not specify functions.

    Some formulas follow. Which specify a function? For those that do specify a function, what is the ``natural'' domain and range (see page 6 just below the red line)? On some problems, a graph might help. You can get one painlessly with as follows:

    Plot[x + 5, {x, -5, 5}]

    Plot[Sin[x] + Cos[x], {x, -2Pi, 2Pi}]

    and so forth.

    1. y = x + 5
    2. y = sin x + cos x
    3. y = 1/x
    4. y - x - 5 = 0
    5. x2 + y2 - 4 = 0
    6. x2 + y2 + 4 = 0

  2. Functions can sometimes be specified by tables.
    1. The ``Denver Diary'' for July 15, 1991 was:

      Time Temp Time Temp
      Mdnt 72 Noon 87
      1 am 69 1 pm 92
      2 am 67 2 pm 93
      3 am 63 3 pm 95
      4 am 61 4 pm 96
      5 am 58 5 pm 97
      6 am 57 6 pm 96
      7 am 64 7 pm 86
      8 am 70 8 pm 81
      9 am 76 9 pm  
      10am 80 10pm  
      11am 84 11pm  



      First, if you are new to Colorado, note the temperature range in a single day! Next: Is Temp a function of Time? Is Time a function of Temp? If you answered ``yes'' to either or both questions, what are the domains and ranges?

    2. The ``Never on Sunday'' data set is:

      Day Answer
      Sunday No
      Monday Yes
      Tuesday Yes
      Wednesday Yes
      Thursday Yes
      Friday Yes
      Saturday Yes



      Is Answer a function of Day? Is Day a function of Answer? If you answered ``yes'' to either or both questions, what are the domains and ranges?

  3. Functions can sometimes be specified by giving (domain, range) pairs.
    1. Is this set of pairs a function: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} If you said ``yes'', domain? range?
    2. Is this set of pairs a function: {(1, 1), (2, 2), (2, 3), (4, 4), (1, 5)} If you said ``yes'', domain? range?

  4. Functions can sometimes be specified by a graph. Which of the following represent a function y = f (x)? Is it feasible to get the domain and range from the graph?
    Figure: Graph a
    \begin{figure} \epsfysize 100pt \centerline{\epsffile{ws2p4a.eps}} \end{figure}

    Figure: Graph b
    \begin{figure} \epsfysize 100pt \centerline{\epsffile{ws2p4b.eps}} \end{figure}

  5. Functions can sometimes be specified in words.
    1. Does ``the straight line passing through the points (1,3), (6, 12)'' describe a function?
    2. What about ``the circle with center (3,4) and radius 5''?
    3. How about ``the monthly population of Vail, Colorado between 1950 and the present''?

  6. Well, even with all these alternatives, it can't be denied that a formula is a useful thing. So now, I'll ask you for some!

    1. A straight line is determined by two points, right? So if the points are P1(x1, y1) and P2(x2, y2), quick now, write a formula for the straight line. Any special cases to note?

    2. A straight line is also determined by a point and a slope. So if the point is P1(x1, y1) and the slope is m, write a formula for the straight line.

    3. A straight line is also determined by its slope and ``intercept''. But isn't this just a special case of the previous case? Anyway, if the slope is m and the y-intercept is b, write down a formula for the straight line.