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≥3f(x) = \begin{cases} x+1 & x < 3 \\ 2x & x \ge 3 \end{cases}, find f(3)f(3).

Example 2

medium
For f(x)={−xx<0xx≥0f(x)=\begin{cases}-x & x<0\\ x & x\ge0\end{cases}, simplify f(x)f(x).

Example 3

medium
Is f(x)={x+1x<12xx≥1f(x)=\begin{cases}x+1 & x<1\\ 2x & x\ge1\end{cases} continuous at x=1x=1?

Example 4

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

Example 5

challenge
Choose aa so that f(x)={ax+1x<2x2x≥2f(x)=\begin{cases}ax+1 & x<2\\ x^2 & x\ge2\end{cases} is continuous at x=2x=2.

Example 6

medium
Find the domain region where f(x)={1/xx<0xx≥0f(x)=\begin{cases}1/x & x<0\\ \sqrt{x} & x\ge0\end{cases} uses x\sqrt{x}.

Example 7

easy
For f(x)={x2x<2x+1x≥2f(x)=\begin{cases}x^2 & x<2\\ x+1 & x\ge2\end{cases}, find f(1)f(1).

Example 8

medium
For f(x)={2x<0x2x≥0f(x)=\begin{cases}2 & x<0\\ x^2 & x\ge0\end{cases}, find f(−5)+f(2)f(-5)+f(2).

Example 9

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

Example 10

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

Example 11

medium
Express f(x)=∣x∣+xf(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|x| + |x - 2| = 6.

Example 14

easy
Evaluate ∣x−2∣|x - 2| at x=−1x = -1.

Example 15

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

Example 16

medium
Is f(x)={x+1x<12x+3x≥1f(x)=\begin{cases}x+1 & x<1\\ 2x+3 & x\ge1\end{cases} continuous at x=1x=1?

Example 17

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

Example 18

hard
Express f(x)=∣x2−1∣f(x) = |x^2 - 1| as a piecewise function.

Example 19

easy
Where does ∣x∣|x| change formula?

Example 20

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