Questions on Piecewise Functions

What's On This Page

This page contains sample problems on piecewise functions. They are for Self-assessment and Review.

Each problem (or group of problems) has an "answer button" ANSWER which you can click to look at an answer. Some solutions have a "further explanation button" Explain which you can click to see a more complete, detailed solution.

What to Do

To gain the most benefit from these problems,

Work the problems on your own. Write down your solutions BEFORE looking at any answers.

Use a graphing calculator as appropriate. A graphing calculator can be used to verify that your answers "make sense" or "look right". Special care needs to be taken in for piecewise functions as explained in the exposition.

If you have difficulties with this material, please contact your instructor. (See Getting Help in Stage 1.) Being able to understand the notation and make accurate graphs of piecewise functions is important to success in CalculusQuestTM.

1. Graph the following piecewise functions and evaluate for the given values of x.
Note: " Evaluate the function f for x = a" means the same as "find f(a)".

  1. { -x3+ 6 x2 - 9 x + 4 for x < 2
    f(x) =
    x - 4 for x 2

  2. { -x2 + 2 for x < -2
    f(x) = 2x + 1 for -2 x < 0
    x 2 + 2 for x 0


    2. Let f(x) = |x|/x.

    1. What is the domain of f?

    2. Evaluate the following: f(-3), f(-21), f(5), f(9).

    3. Draw the graph of f(x).

    4. This function is called the signum function and is usually written sgn(x). Rewrite the rule for sgn(x) using piecewise notation.


    3. 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 $35 02.50 + 28%$23,350
    $56,550$117,950 $12,798.50 + 31%$56,550

    1. 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.

    2. Evaluate the function to find the tax owed by a single person whose taxable income in 1995 was $31,950.


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