Practice Restricted Domain in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

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

Example 2

easy

What is the standard restricted domain of

\sin x

for defining

\arcsin

Example 3

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 4

medium

Find the natural domain of

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

Example 5

hard

Find the natural domain of

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

Example 6

medium

Find the natural domain of

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

Example 7

medium

A function is increasing on

[0,2]

and decreasing on

[2,4]

. On which subinterval can you restrict for invertibility?

Example 8

challenge

Restrict

f(x) = \sin x

to make it invertible on an interval containing

x = \pi

, and give the interval.

Example 9

medium

Find the domain of

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

Example 10

hard

Find the natural domain of

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

Example 11

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 12

medium

Find the inverse of

f(x) = x^2

restricted to

x \ge 0

Example 13

medium

For

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

, find the natural domain restriction.

Example 14

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 15

medium

Find the natural domain of

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

Example 16

easy

Find the natural domain of

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

Example 17

medium

Restrict

f(x) = \tan x

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

Example 18

easy

On what restricted domain is

f(x) = x^2

one-to-one and increasing?

Example 19

medium

Find the domain of

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

Example 20

easy

After restricting

f(x) = x^2

x \ge 0

, what is the range?

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Related Concepts

Domain Function Inverse Function