Practice Piecewise Behavior in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Piecewise behavior refers to a function that exhibits qualitatively different characteristics in different regions of its domain, like having a different slope or curvature in each region.

Think of the behavior as shifting gears — the function follows one rule until it hits a boundary, then switches to a different rule for the next region.

Showing a random 20 of 50 problems.

Example 1

easy
For f(x)={x+1x<32xx≥3, find f(3).

Example 2

medium
For f(x)={−xx<0xx≥0, simplify f(x).

Example 3

medium
Is f(x)={x+1x<12xx≥1 continuous at x=1?

Example 4

easy
How many rules does a 3-piece function use?

Example 5

challenge
Choose a so that f(x)={ax+1x<2x2x≥2 is continuous at x=2.

Example 6

medium
Find the domain region where f(x)={1/xx<0xx≥0 uses x.

Example 7

easy
For f(x)={x2x<2x+1x≥2, find f(1).

Example 8

medium
For f(x)={2x<0x2x≥0, find f(−5)+f(2).

Example 9

easy
Write ∣2x+1∣ as a piecewise function (no absolute value).

Example 10

challenge
Express f(x)=max⁡(x,3−x) as a piecewise function and find its minimum.

Example 11

medium
Express f(x)=∣x∣+x as a piecewise function.

Example 12

easy
Does each region of a piecewise function need the same kind of formula?

Example 13

challenge
Solve ∣x∣+∣x−2∣=6.

Example 14

easy
Evaluate ∣x−2∣ at x=−1.

Example 15

hard
For f(x)=∣x∣+∣x−4∣, find the MINIMUM value of f and where it occurs.

Example 16

medium
Is f(x)={x+1x<12x+3x≥1 continuous at x=1?

Example 17

medium
Solve ∣x+2∣≤4 and express as an interval.

Example 18

hard
Express f(x)=∣x2−1∣ as a piecewise function.

Example 19

easy
Where does ∣x∣ change formula?

Example 20

hard
Sketch the rate of change of f(x)=∣x−3∣. Where is the rate +1, and where is it −1?
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