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Piecewise and Absolute Value Functions

A free College Algebra lesson from the “Advanced Function Types” unit, with a worked example and practice problems including step-by-step solutions.

Piecewise functions use different rules on different input intervals. Absolute value functions form V-shaped graphs and measure distance from a center.

What you'll learn

Why it matters: Tax brackets, shipping rates, parking fees, and distance-from-center models all change rules depending on the input. Piecewise notation teaches students to choose the rule first, then calculate.

Worked example

Problem. If f(x) = |x - 4|, find f(10).

  1. Substitute 10.
  2. |10 - 4| = |6|.
  3. The output is 6.

Answer: 6

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. If f(x) = |x + 3|, find f(2).

Show solution
  1. |2 + 3| = 5.

Answer: 5

2. The vertex of y = |x - 5| + 2 is...

Choices: (5, 2) · (-5, 2) · (5, -2) · (-5, -2)

Show solution
  1. Use (h, k).

Answer: (5, 2)

3. A piecewise function may use...

Choices: Different rules on different intervals · Only one constant · No input values · Only circles

Show solution
  1. That is the purpose of piecewise notation.

Answer: Different rules on different intervals

4. If f(x) = 2x for x < 0 and f(x) = x + 3 for x >= 0, find f(4).

Show solution
  1. Use the x >= 0 rule.

Answer: 7

5. The vertex of y = |x - 5| is...

Choices: (5, 0) · (-5, 0) · (0, 5) · (0, -5)

Show solution
  1. Set the expression inside absolute value equal to 0.

Answer: (5, 0)

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