Restricted Domain Examples in Math

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

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

Concept Recap

Restricting a domain limits allowable inputs so a function has desired properties, often invertibility.

You keep only the input interval where the function behaves one way.

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: Restricting a domain throws away part of the input set so a function gains a property like invertibility.

Common stuck point: The procedure for restricted domain is the easy part; the trap is restricting to an interval where the property still fails. Asking "Am I deliberately limiting the inputs so the function gains a property (like passing the horizontal line test)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I deliberately limiting the inputs so the function gains a property (like passing the horizontal line test)?

Worked Examples

Example 1

easy

Find the natural (implied) domain of

f(x) = \sqrt{x - 3}

Answer

[3, \infty)

First step

The square root function requires a non-negative argument:

x - 3 \ge 0

Full solution

2
Solve: $x \ge 3$ .
3
The domain is $[3, \infty)$ .

A restricted domain limits the inputs of a function to a subset of all real numbers. For square roots, the expression under the radical must be non-negative. For fractions, the denominator cannot be zero. These natural restrictions come from the mathematical definition of the operations involved.

Example 2

medium

Restrict the domain of

f(x) = x^2

so that the function has an inverse. Find the inverse on this restricted domain.

Example 3

medium

Find the natural domain of

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

Example 4

medium

Restrict

f(x) = x^2 - 4x + 7

x \ge 2

and find the inverse.

Example 5

hard

Find the largest restricted domain of

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

on which the function is invertible, containing

x = 0

Example 6

hard

Restrict the domain of

f(x) = \cos x

to make it invertible on an interval containing

x = 2\pi

, then state the inverse's range on that branch.

Example 7

challenge

Restrict the domain of

f(x) = x^2 e^x

to the largest interval where

f

is one-to-one and contains

x = 2

Practice Problems

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

Example 1

medium

Find the domain of

g(x) = \frac{\sqrt{x+4}}{x^2 - 9}

Example 2

hard

Restrict the domain of

f(x) = (x - 1)^2 + 3

to the largest interval containing

x = 4

on which

f

is one-to-one, then find

f^{-1}(x)

Example 3

easy

On what restricted domain is

f(x) = x^2

one-to-one and increasing?

Example 4

easy

What is the standard restricted domain of

\sin x

for defining

\arcsin

Example 5

easy

To invert

f(x) = x^2

giving the negative square root branch, restrict to what domain?

Example 6

easy

f(x) = x^3

one-to-one without any restriction?

Example 7

easy

After restricting

f(x) = x^2

x \ge 0

, what is the range?

Example 8

easy

Which should you restrict to make a function invertible: the domain or the range?

Example 9

easy

On what domain is

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

one-to-one and increasing?

Example 10

easy

The restricted domain of

\cos x

for

\arccos

is which interval?

Example 11

medium

Restrict

f(x) = x^2 - 6x + 5

to make it invertible with increasing branch, and give the interval.

Example 12

medium

Find the inverse of

f(x) = x^2

restricted to

x \ge 0

Example 13

medium

f(x) = x^2

restricted to

[-1, 2]

one-to-one?

Example 14

medium

For

f(x) = \sqrt{x-3}

, find the natural domain restriction.

Example 15

medium

Restrict

f(x) = \tan x

to a one-to-one interval for its inverse.

Example 16

medium

A function is increasing on

[0,2]

and decreasing on

[2,4]

. On which subinterval can you restrict for invertibility?

Example 17

medium

After restricting

f(x) = x^2

x \le 0

, what is the domain of

f^{-1}

Example 18

medium

Why must you update the range when restricting a domain?

Example 19

medium

For

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

, find the natural domain restriction.

Example 20

challenge

Find the largest interval containing

x=1

on which

f(x) = x^2 - 4x + 3

is one-to-one.

Example 21

challenge

Restrict

f(x) = \sin x

to make it invertible on an interval containing

x = \pi

, and give the interval.

Example 22

challenge

For

f(x) = x^2

restricted to

[a, a+2]

, find all

a

making

f

one-to-one.

Example 23

easy

Find the natural domain of

f(x) = \sqrt{5 - x}

Example 24

easy

Find the natural domain of

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

Example 25

easy

Restrict the domain of

f(x) = (x + 3)^2

to make it one-to-one and decreasing.

Example 26

medium

Find the natural domain of

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

Example 27

medium

Restrict the domain of

f(x) = x^2 - 4x + 7

so that the function has an inverse with range containing all

x \ge 3

Example 28

medium

Find the domain of

f(x) = \dfrac{1}{\sqrt{x^2 - 9}}

Example 29

medium

Find the domain of

f(x) = \arcsin(2x - 1)

Example 30

medium

Find the natural domain of

g(x) = \sqrt{x} + \sqrt{4 - x}

Example 31

medium

Find the natural domain of

f(x) = \dfrac{x^2}{x^2 - 5x + 6}

Example 32

hard

Find the natural domain of

h(x) = \ln(x^2 - 4x + 3)

Example 33

hard

Find the natural domain of

f(x) = \sqrt{\sin x}

[0, 2\pi]

Example 34

hard

Find the natural domain of

f(x) = \dfrac{1}{\ln x}

Example 35

hard

Find the natural domain of

g(x) = \arccos\!\left(\dfrac{x}{x+1}\right)

Example 36

hard

Find the natural domain of

f(x) = \sqrt{x - 1} + \ln(5 - x)

Example 37

challenge

Find the natural domain of

f(x) = \dfrac{\sqrt{4 - x^2}}{\ln(x + 1)}

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Related Concepts

Domain Function Inverse Function

Background Knowledge

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

domainfunction definitioninverse function