Answers to Questions on Piecewise Functions


    1. Question: Graph the following piecewise function and evaluate for the given values of x.
      { -x3+ 6 x2 - 9 x + 4 for x < 2
      f(x) =
      x - 4 for x 2

      Answer:

      f(-1) = 20 (Use the first formula since -1 < 2.)
      f(0) = 4 (Use first formula again.)
      f(2) = -2 (Use the second formula.)
      f(4) = 0 (Second formula)

    2. Question: Graph the following piecewise function and evaluate for the given values of x.
      { -x2 + 2 for x < -2
      f(x) = 2x + 1 for -2 x < 0
      x 2 + 2 for x 0

      Answer:
      f(5) = 27 (Use the third piece formula.)
      f(-2) = -3 (Use the second formula.)
      f(-4) = -14

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    3. Let f(x) = |x|/x.
      1. Question: What is the domain of f?
        Answer: The domain of f is all real numbers except 0.

      2. Question: Evaluate the following: f(-3), f(-21), f(5), f(9).
        Answer:
        f(-3) = -1
        f(-21) = -1
        f(5) = 1
        f(9) = 1.

      3. Question: Draw the graph of f(x).

      4. Question: This function is called the signum function and is usually written sgn(x). Rewrite the rule for sgn(x) using piecewise notation.
        Answer:
        { -1 for x < 0
        f(x) =
        1 for x > 0

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    4. In the 1995 tax form a tax rate schedule is given for people whose filing status is single. Part of the table is shown below:

      If the taxable income is over...But not over--then the tax is...of the amount over--
      $0$23,350 15% $0
      $23,350$56,550 $3,502.50 + 28%$23,350
      $56,550$117,950 $12,798.50 + 31%$56,550

      1. Question: Write the defining rule for a piecewise function T(x) giving the tax owed by a person whose taxable income is $x, where x is less than $117,950.
        Answer:
        { 0.15x for x < 23,350
        f(x) = 3502.5 + 0.28(x - 23,350) for 23,350 x < 56,550
        12,798.5 + 0.31(x - 56,550) for 56,550 > x 117,950

      2. Question: Evaluate the function to find the tax owed by a single person whose taxable income in 1995 was $31,950.
        Answer: $5,910.50.
        Explain

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