More Operations with Functions: Practice Exercises

Here are various Exercises to accompany the section More Operations with Functions.  For this section, you should assume the following function definitions:

  1. Let f be given by the following table:
    a f(a)
    -2
    -1
    0
    1
    2
    3
    4
    5    		 1
    -2
    4
    5
    -1
    0
    3
    2
  2. Let g(x) = 3x + 2.
  3. Let h be given by the following arrow diagram:

  4. Let p be the function which "squares the input, then subtracts 1 from the result."
  5. Let q be given by the following set of ordered pairs: {(-2, 3), (-1, 1), (0, -2), (1, 0), (2, 4), (3, 5), (4, -1), (5, 2)}.
  6. Let s(b) = (b - 2)/3.
  7. Let k be given by the following arrow diagram:

  8. Let t be the function which "takes the reciprocal of the input, then adds 1 to the result."
  9. Let w be given by the following table:
    x w(x)
    -2
    -1
    0
    1
    2
    3
    4
    5    		 -1
    0
    3
    2
    -2
    4
    5
    1

Arithmetic with Functions

  1. Using the definitions given above, evaluate each of the following functions at the given input value:
    1. (f + w)(4)
    2. (s - g)(2)
    3. (h · k)(3)
    4. (p/q)(2)
    5. (3t)(5)

    Solution.

  2. Using the definitions given above, compute the indicated description for each of the following functions:
    1. Give an arrow diagram for h · k.  Hint: Compute the value of the product on each possible input, then draw a picture of the result.
    2. Give the table of values for f + w.  Hint: Compute the value of the sum on each possible input.
    3. Give a formula for p - g.  Hint: First determine the formula for p.
    4. Give a formula for s/g.  
    5. Give a formula for 3t.  Hint: First determine the formula for t.

    Solution.

Joining Functions

  1. Consider the following piecewise-defined function:

     

    Compute the following:

    1. f(-2)
    2. f(-1)
    3. f(0)
    4. f(1)
    5. f(2)
    6. f(3)
    7. f(4)

    Solution.

Decomposing Functions

  1. Express each of following functions in the indicated manner:
    1. If , determine formulas for functions g and k, so that , if h is the square root function.
    2. If f(x) = 4(-x + 3)2 - 5, determine formulas for functions g and k, so that , if h is the square function.
    3. If , determine formulas for functions g, h, and k, so that f = k/g + h.
    4. If f(x) = x2(2x + 1) - |3x - 4|, determine formulas for functions g, h, and k, so that f = k·g - h.

    Solution.

Go to Principles of Graphing .

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