Piecewise Function Examples in Math

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

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

Concept Recap

A piecewise function is defined by different formulas on different non-overlapping intervals of its domain, with the applicable formula determined by the input value.

A piecewise function is like a rulebook: look up which rule applies to your input value, then use only that rule to compute the output.

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 piecewise function uses different formulas on different input intervals; the input decides which one applies.

Common stuck point: The procedure for piecewise function is the easy part; the trap is evaluating with the wrong piece. Asking "Does the formula used depend on which interval the input falls into?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the formula used depend on which interval the input falls into?

Worked Examples

Example 1

easy

Evaluate

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

x = -3

x = 0

, and

x = 4

Answer

f(-3)=9,\; f(0)=1,\; f(4)=9

First step

For

x = -3

: since

-3 < 0

, use

f(x) = x^2

. So

f(-3) = (-3)^2 = 9

Full solution

2
For $x = 0$ : since $0 \geq 0$ , use $f(x) = 2x+1$ . So $f(0) = 2(0)+1 = 1$ .
3
For $x = 4$ : since $4 \geq 0$ , use $f(x) = 2x+1$ . So $f(4) = 2(4)+1 = 9$ .

In a piecewise function, the domain is partitioned into intervals, each with its own rule. The key skill is identifying which interval the input belongs to before applying the corresponding formula.

Example 2

medium

Determine whether

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

is continuous at

x = 2

Example 3

medium

Find the value of

a

that makes

f(x) = \begin{cases} ax + 1 & x \le 2 \\ x^2 - 1 & x > 2 \end{cases}

continuous at

x = 2

Example 4

medium

Determine whether

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

is continuous at

x = 1

Example 5

medium

For what value of

k

f(x) = \begin{cases} kx + 1 & x \le 1 \\ 3x - 1 & x > 1 \end{cases}

continuous at

x = 1

Example 6

medium

A shipping company charges $5 for packages up to

2

lb, $8 for packages over

2

lb but at most

10

lb, and $12 for packages over

10

lb. Write the cost

C(w)

as a piecewise function.

Example 7

hard

Find

a

and

b

so that

f(x) = \begin{cases} ax + b & x < 1 \\ x^2 & 1 \le x \le 2 \\ bx + 2 & x > 2 \end{cases}

is continuous everywhere.

Example 8

hard

The greatest integer (floor) function is

\lfloor x \rfloor

. Express

\lfloor x \rfloor

[0, 3)

as a piecewise function.

Example 9

hard

Find all values of

c

such that

f(x) = \begin{cases} cx^2 + 1 & x \le 1 \\ 2cx + 3 & x > 1 \end{cases}

is continuous at

x = 1

Example 10

challenge

Find

a

and

b

so that

f(x) = \begin{cases} x^2 + a & x \le 1 \\ bx + 2 & 1 < x < 3 \\ 5 & x \ge 3 \end{cases}

is continuous everywhere.

Practice Problems

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

Example 1

easy

Given

g(x) = \begin{cases} -x & x < -1 \\ x^2 & -1 \leq x \leq 2 \\ 5 & x > 2 \end{cases}

, evaluate

g(-2)

g(1)

, and

g(3)

Example 2

hard

Find the value of

c

so that

f(x) = \begin{cases} cx + 1 & x \leq 3 \\ x^2 - 2 & x > 3 \end{cases}

is continuous at

x = 3

Example 3

easy

For

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

, find

f(-2)

Example 4

easy

For the same

f

, find

f(3)

Example 5

easy

For

f(x)=\begin{cases}5 & x\le 1\\ 2x & x>1\end{cases}

, find

f(1)

Example 6

easy

For the same

f

, find

f(4)

Example 7

easy

In the absolute value

|x|=\begin{cases}x & x\ge 0\\ -x & x<0\end{cases}

, find

|-5|

Example 8

easy

How many pieces does

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

have?

Example 9

easy

Should the intervals of a piecewise function overlap?

Example 10

easy

For

f(x)=\begin{cases}x^2 & x\le 2\\ 10 & x>2\end{cases}

, find

f(2)

Example 11

medium

For

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

, is

f

continuous at

x=2

Example 12

medium

For

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

, is

f

continuous at

x=1

Example 13

medium

Evaluate

f(-1)+f(2)

for

f(x)=\begin{cases}3x & x<0\\ x^2 & x\ge 0\end{cases}

Example 14

medium

A taxi charges $3 for the first mile and $2 per mile after. Write the cost for

m>1

miles.

Example 15

medium

For

f(x)=\begin{cases}-x & x<0\\ x^2 & 0\le x\le 2\\ 4 & x>2\end{cases}

, find

f(0)

and

f(2)

Example 16

medium

The step function

f(x)=\lfloor x\rfloor

(greatest integer). Find

f(2.7)

and

f(-1.2)

Example 17

medium

Find the domain of

f(x)=\begin{cases}\sqrt{x} & 0\le x\le 4\\ 8-x & x>4\end{cases}

Example 18

medium

Does

f(x)=\begin{cases}1 & x\le 0\\ x+1 & x>0\end{cases}

have a jump at

x=0

Example 19

medium

For

f(x)=\begin{cases}x^2 & x<0\\ 5 & x=0\\ 2x & x>0\end{cases}

, find

f(0)

Example 20

challenge

Find

a

so that

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

is continuous at

x=2

Example 21

challenge

Express

f(x)=|x-3|

as a piecewise function and find

f(1)

Example 22

challenge

For

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

, find all

x

with

f(x)=1

Example 23

easy

For

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

, find

f(0)

Example 24

easy

For the same

f

above, find

f(5)

Example 25

easy

A function is

f(x) = 1

x < 0

f(x) = 2

0 \le x < 5

, and

f(x) = 3

x \ge 5

. Find

f(5)

Example 26

easy

For

f(x) = \begin{cases} x + 2 & x \le 0 \\ 3x & x > 0 \end{cases}

, find

f(0)

Example 27

medium

A taxi charges $3 base plus $0.50 per mile for the first

5

miles and $0.30 per additional mile. Write the fare as a piecewise function of miles

m

Example 28

medium

For

f(x) = \begin{cases} x + 3 & x < 0 \\ x^2 & 0 \le x \le 3 \\ 9 & x > 3 \end{cases}

, find

f(-1) + f(2) + f(4)

Example 29

medium

Solve

f(x) = 4

for

f(x) = \begin{cases} 2x + 6 & x < 0 \\ x^2 & x \ge 0 \end{cases}

Example 30

medium

Sketch (describe) the graph of

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

Example 31

medium

For

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

, find the limits

\lim_{x \to 1^-}

and

\lim_{x \to 1^+}

Example 32

hard

Solve

f(x) = 5

for

f(x) = \begin{cases} x + 2 & x < 3 \\ x^2 - 4 & x \ge 3 \end{cases}

Example 33

hard

For

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

, find the range on

[-2, 2]

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Related Concepts

Function Domain Absolute Value Step Function Intuition

Background Knowledge

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

function definitiondomain