Functions and Algebra: Practice Exercises

Here are various Exercises to accompany the section Functions and Algebra.  For the first two sets of Exercises in 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 "cubes the input, then subtracts 3 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 r be given by the following plot:

  7. Let s(b) = (b - 2)/3.
  8. Let k be given by the following arrow diagram:

  9. Let t be the function which "takes the reciprocal of the input, then adds 1 to the result."
  10. 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

Evaluation and Composition

  1. Using the definitions given above, evaluate each of the following composite functions at the given input value:

    Solution.

  2. Compute the indicated description for each of the following composite functions:
    1. Give a verbal description of .
    2. Give a formula for .  Hint: First determine the formula for p.
    3. Give the table of values for .  Hint: Compute the value of the composite on each possible input.
    4. Give an arrow diagram for .  Hint: Compute the value of the composite on each possible input, then draw a picture of the result.
    5. Give a formula for .

    Solution.

Solving and Inverse Functions

  1. Using the definitions given above, give all possible solutions to each of following equations:
    1. f(x) = 5.
    2. h(x) = 2.
    3. p(x) = 24.
    4. r(x) = 4.
    5. q(x) = -1.
    6. w(x) = 3.

    Solution.

  2. Using the definitions given above, compute the reverse in the indicated form of each of following functions:

    1. The reverse of f as a table.
    2. The reverse of h as an arrow diagram.
    3. The reverse of p in words.
    4. The reverse of r as a plot.  Hint: First make a table of 4 - 6 points, reverse the table, then plot.
    5. The reverse of q as a table.
    6. The reverse of w as an arrow diagram.
    7. The reverse of t as a formula.

    Solution.

  3. Explain why each of the functions from the previous Exercise are or are not invertible.

    1. f.
    2. h.
    3. p.
    4. r.
    5. q.
    6. w.
    7. t.

    Solution.

  4. This Exercise is designed to help you see why "division by 0" is "not allowed" in Algebra.

    1. Let s be the "multiply by 0" function.  Complete the following arrow diagram for s:

    2. How would you describe the "opposite" of s in words? 
    3. As in the previous Exercise, explain why s is or is not invertible.

    Solution.

More on Inverse Functions

  1. Following the example in the text, and using the definitions given above, give all possible solutions to each of following equations. 
    1. 2p(x - 1) - 3 = 7.
    2. (-1/2)f(3x - 1) + 5 = 3.
    3. 5t(2 - x) + 3 = 3.
    4. -3h(x - 1) + 1 = -5.  Note: Since is not invertible, you will get more than one solution.

    Solution.

  2. For each of following functions, use the Theorem in text to compute its inverse.  Verify each answer by showing that it satisfies the cancellation equations. 
    1. If g(x) = 3x + 2, compute g -1.
    2. If p(x) = x3 - 3, compute p -1.
    3. If f(x) = 1/(x + 4), compute f -1.

    Solution.

  3. Solve each of following equations for x.  Leave your answer as a formula in y, involving an appropriate inverse function. 
    1. Solve 2p(x - 1) - 3 = y for x, in terms of p -1.
    2. (-1/2)f(3x - 1) + 5 = y for x, in terms of f -1.
    3. 5t(2 - x) + 3 = y for x, in terms of t -1.

    Solution.

Go to More Operations with Functions .

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