Piecewise Behavior Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Piecewise Behavior.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A function shows piecewise behavior when it follows one rule until a boundary, then switches to a qualitatively different rule.

Common stuck point: The procedure for piecewise behavior is the easy part; the trap is assuming continuity for free. Asking "Does the function switch to a different rule depending on which region of the domain the input is in?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the function switch to a different rule depending on which region of the domain the input is in?

Worked Examples

Example 1

easy

Write

|x|

as an explicit piecewise function, evaluate

|-4|

|0|

, and

|7|

, and sketch its graph.

Answer

|-4|=4

|0|=0

|7|=7

; V-shaped graph with vertex at origin

First step

Definition:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Full solution

2
Evaluate: $|-4| = -(-4) = 4$ ; $|0| = 0$ ; $|7| = 7$ .
3
Graph: two rays meeting at the origin $(0,0)$ , slope $-1$ for $x<0$ and slope $+1$ for $x\geq0$ , forming a 'V' shape.

The absolute value function is the simplest and most important piecewise function. It measures distance from zero, always returning a non-negative value. Its V-shape has slope

\pm1

and vertex at the origin.

Example 2

medium

Solve the equation

|2x - 5| = 7

and the inequality

|2x - 5| < 7

Example 3

medium

Solve

|3x - 6| = 9

Example 4

medium

Solve

|x + 2| \le 4

and express as an interval.

Example 5

hard

Express

f(x) = |x^2 - 1|

as a piecewise function.

Example 6

hard

Find constants

a

and

b

so that

f(x) = \begin{cases} x^2 & x < 2 \\ ax + b & x \ge 2 \end{cases}

is BOTH continuous AND has a matching slope at

x = 2

Example 7

challenge

Express

f(x) = \max(x, 3 - x)

as a piecewise function and find its minimum.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Simplify

|x-3| + |x+1|

for

x = -2

x = 1

, and

x = 5

Example 2

hard

Express

f(x) = |x^2 - 4|

as a piecewise function (without absolute value bars) and identify where

f(x) = 0

Example 3

easy

For

f(x)=\begin{cases}x & x<0\\ 2x & x\ge0\end{cases}

, find

f(-3)

Example 4

easy

For the same

f

, find

f(4)

Example 5

easy

Which piece applies to

x=0

f(x)=\begin{cases}1 & x<0\\ 5 & x\ge0\end{cases}

Example 6

easy

Does a piecewise function have to be continuous?

Example 7

easy

How many rules does a 3-piece function use?

Example 8

easy

For

f(x)=\begin{cases}x^2 & x<2\\ x+1 & x\ge2\end{cases}

, find

f(1)

Example 9

easy

For the same

f

, find

f(2)

Example 10

easy

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

Example 11

medium

f(x)=\begin{cases}x+1 & x<1\\ 2x & x\ge1\end{cases}

continuous at

x=1

Example 12

medium

f(x)=\begin{cases}x+1 & x<1\\ 2x+3 & x\ge1\end{cases}

continuous at

x=1

Example 13

medium

For

f(x)=\begin{cases}-x & x<0\\ x & x\ge0\end{cases}

, simplify

f(x)

Example 14

medium

Evaluate

f(-2)+f(3)

for

f(x)=\begin{cases}x^2 & x<0\\ 2x-1 & x\ge0\end{cases}

Example 15

medium

A function is

f(x)=x^2

for

x\le1

and a line for

x>1

. If the line continues the value smoothly with

f(1)=1

, what is

f(1)

Example 16

medium

For

f(x)=\begin{cases}3 & x<2\\ x+1 & x\ge2\end{cases}

, on which region is

f

constant?

Example 17

medium

For

f(x)=\begin{cases}2x & x<3\\ x^2 & x\ge3\end{cases}

, is the rate constant on

x<3

Example 18

medium

Find the domain region where

f(x)=\begin{cases}1/x & x<0\\ \sqrt{x} & x\ge0\end{cases}

uses

\sqrt{x}

Example 19

challenge

Choose

a

so that

f(x)=\begin{cases}ax+1 & x<2\\ x^2 & x\ge2\end{cases}

is continuous at

x=2

Example 20

challenge

For

f(x)=\begin{cases}x+k & x<1\\ 3x & x\ge1\end{cases}

, find

k

making it continuous, then compute

f(0)

Example 21

challenge

A piecewise function is

x^2

for

x<0

0

for

0\le x\le1

, and

x-1

for

x>1

. Where is it NOT continuous?

Example 22

medium

For

f(x)=\begin{cases}2 & x<0\\ x^2 & x\ge0\end{cases}

, find

f(-5)+f(2)

Example 23

easy

Evaluate

|x - 2|

x = 5

Example 24

easy

Evaluate

|x - 2|

x = -1

Example 25

easy

Write

|2x + 1|

as a piecewise function (no absolute value).

Example 26

easy

For

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

, find

f(3)

Example 27

easy

Find

|x| + |x - 3|

when

x = 1

Example 28

medium

For

f(x) = \begin{cases} 1-x & x < 0 \\ x^2 & 0 \le x \le 2 \\ 3 & x > 2 \end{cases}

, compute

f(-1) + f(2) + f(5)

Example 29

medium

f(x) = \begin{cases} x^2 & x < 1 \\ 2x - 1 & x \ge 1 \end{cases}

continuous at

x = 1

Example 30

medium

Find

a

so that

f(x) = \begin{cases} ax^2 & x < 1 \\ 2x + 1 & x \ge 1 \end{cases}

is continuous at

x = 1

Example 31

medium

Find

\lfloor -2.5 \rfloor

(floor of

-2.5

Example 32

medium

Express

f(x) = |x| + x

as a piecewise function.

Example 33

medium

For what

x

does

|x - 4|

equal

|x|

Example 34

hard

For

f(x) = |x| + |x - 4|

, find the MINIMUM value of

f

and where it occurs.

Example 35

hard

For

f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 + 1 & x \ge 0 \end{cases}

, sketch its key features and identify any jump at

x = 0

Example 36

hard

Solve

|x - 1| = 2x - 5

(be careful — check both branches and the validity).

Example 37

hard

Sketch the rate of change of

f(x) = |x - 3|

. Where is the rate

+1

, and where is it

-1

Example 38

challenge

For

f(x) = \lfloor x \rfloor

, find all

x

where

f(x) = 2

Example 39

challenge

Solve

|x| + |x - 2| = 6

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Related Concepts

Piecewise Function Absolute Value

Background Knowledge

These ideas may be useful before you work through the harder examples.

piecewise function