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
- Interpret piecewise rules
- Evaluate absolute value functions
- Connect graph shape to rules
Worked example
Problem. If f(x) = |x - 4|, find f(10).
- Substitute 10.
- |10 - 4| = |6|.
- The output is 6.
Answer: 6
Practice problems
1. If f(x) = |x + 3|, find f(2).
Show solution
- Warm-up: First identify exactly what the question is asking: If f(x) = |x + 3|, find f(2).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- |2 + 3| = 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
2. The vertex of y = |x - 5| + 2 is...
Choices: (5, 2) · (-5, 2) · (5, -2) · (-5, -2)
Show solution
- Core Practice: First identify exactly what the question is asking: The vertex of y = |x - 5| + 2 is...
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- Use (h, k).
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Challenge: First identify exactly what the question is asking: A piecewise function may use...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- That is the purpose of piecewise notation.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Piecewise Evaluation: First identify exactly what the question is asking: If f(x) = 2x for x < 0 and f(x) = x + 3 for x >= 0, find f(4).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Use the x >= 0 rule.
- Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.
Answer: 7
5. The vertex of y = |x - 5| is...
Choices: (5, 0) · (-5, 0) · (0, 5) · (0, -5)
Show solution
- Absolute Value Graphs: First identify exactly what the question is asking: The vertex of y = |x - 5| is...
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- Set the expression inside absolute value equal to 0.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (5, 0)
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