Range Examples in Math

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

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

Concept Recap

The range of a function is the set of all actual output values that the function can produce for inputs in its domain.

The range is the set of all possible "answers" the function can give — some output values may be unreachable no matter what valid input you choose.

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: The range is every value the function can reach as the input runs over the whole domain.

Common stuck point: The procedure for range is the easy part; the trap is copying the domain as the range. Asking "Which output values does the function actually reach as $x$ runs over its domain?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Which output values does the function actually reach as $x$ runs over its domain?

Worked Examples

Example 1

easy

Find the range of

f(x) = x^2 + 1

Answer

[1, \infty)

First step

Since

x^2 \geq 0

for all real

x

, the minimum value of

x^2

0

Full solution

2
Therefore $f(x) = x^2 + 1 \geq 0 + 1 = 1$ .
3
As $x \to \pm\infty$ , $f(x) \to \infty$ , so the range is $[1, \infty)$ .

For quadratics

f(x) = ax^2 + bx + c

with

a > 0

, the minimum value occurs at the vertex. Adding a constant shifts the range upward.

Example 2

medium

Find the range of

g(x) = \frac{2x + 1}{x - 3}

for

x \neq 3

Example 3

medium

Find the range of

f(x)=\sqrt{9-x^2}

Example 4

hard

Find the range of

f(x)=\dfrac{2x^2+1}{x^2+1}

Practice Problems

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

Example 1

easy

Find the range of

f(x) = -|x| + 4

Example 2

hard

Find the range of

f(x) = \frac{x^2 - 4}{x^2 + 1}

Example 3

easy

Find the range of

f(x)=x^2

Example 4

easy

Find the range of

f(x)=x+5

Example 5

easy

Find the range of

f(x)=|x|

Example 6

easy

Find the range of

f(x)=x^2+3

Example 7

easy

Find the range of

f(x)=2^x

Example 8

easy

Find the range of

f(x)=-x^2

Example 9

easy

Find the range of the constant function

f(x)=7

Example 10

easy

Find the range of

f(x)=\sqrt{x}

Example 11

medium

Find the range of

f(x)=x^2-4x+1

Example 12

medium

Find the range of

f(x)=\frac{1}{x}

Example 13

medium

Find the range of

f(x)=3+2\sin x

Example 14

medium

Find the range of

f(x)=\frac{2x+1}{x-1}

Example 15

medium

Find the range of

f(x)=e^x+2

Example 16

medium

Find the range of

f(x)=\sqrt{x}+1

for

x\ge 0

Example 17

medium

Find the range of

f(x)=\frac{x^2}{x^2+1}

Example 18

medium

Find the range of

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

Example 19

challenge

Find the range of

f(x)=\frac{x}{x^2+1}

Example 20

challenge

Find the range of

f(x)=\sin x+\cos x

Example 21

challenge

The function

f(x)=\frac{x^2+1}{x}

for

x>0

has what minimum output?

Example 22

medium

Find the range of

f(x)=5-2^x

Example 23

easy

Find the range of

f(x)=x^2-5

Example 24

easy

Find the range of

f(x)=3x+2

Example 25

easy

Range of

f(x)=-x^2+10

Example 26

easy

Range of

f(x)=e^{-x}

Example 27

easy

Range of

f(x)=\sqrt{x+4}

Example 28

easy

Range of the constant function

f(x)=-2

Example 29

medium

Find the range of

f(x)=x^2-6x+10

Example 30

medium

Find the range of

f(x)=\dfrac{1}{x-2}

Example 31

medium

Find the range of

f(x)=2\cos x-3

Example 32

medium

Find the range of

f(x)=\ln(x-1)

Example 33

medium

Find the range of

f(x)=3-e^x

Example 34

medium

Find the range of

f(x)=\dfrac{x-3}{x+1}

Example 35

medium

Find the range of

f(x)=|x-5|+|x+5|

Example 36

medium

Find the range of

f(x)=x^3

Example 37

medium

Find the range of

f(x)=4-\sqrt{x}

for

x\ge 0

Example 38

hard

Find the range of

f(x)=\dfrac{x^2}{x^2+4}

Example 39

hard

Find the range of

f(x)=2\sin x-3\cos x

Example 40

hard

Find the range of

f(x)=\sqrt{x^2+1}

Example 41

hard

Find the range of

f(x)=\tan x

\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

Example 42

hard

Find the range of

f(x)=x+\dfrac{4}{x}

for

x>0

Example 43

hard

Find the range of

f(x)=\dfrac{1}{1+x^2}

Example 44

challenge

Find the range of

f(x)=\dfrac{x^2-2x+2}{x-1}

for

x>1

← Back to Range Formula Explained Practice Mode

Related Concepts

Function Domain

Background Knowledge

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

function definitiondomain