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 sinโกx\sin x for defining arcsinโก\arcsin?

Example 3

hard
Restrict the domain of f(x)=(xโˆ’1)2+3f(x) = (x - 1)^2 + 3 to the largest interval containing x=4x = 4 on which ff is one-to-one, then find fโˆ’1(x)f^{-1}(x).

Example 4

medium
Find the natural domain of f(x)=xโˆ’1x+2f(x) = \sqrt{\dfrac{x - 1}{x + 2}}.

Example 5

hard
Find the natural domain of h(x)=lnโก(x2โˆ’4x+3)h(x) = \ln(x^2 - 4x + 3).

Example 6

medium
Find the natural domain of g(x)=x+4โˆ’xg(x) = \sqrt{x} + \sqrt{4 - x}.

Example 7

medium
A function is increasing on [0,2][0,2] and decreasing on [2,4][2,4]. On which subinterval can you restrict for invertibility?

Example 8

challenge
Restrict f(x)=sinโกxf(x) = \sin x to make it invertible on an interval containing x=ฯ€x = \pi, and give the interval.

Example 9

medium
Find the domain of f(x)=arcsinโก(2xโˆ’1)f(x) = \arcsin(2x - 1).

Example 10

hard
Find the natural domain of f(x)=1lnโกxf(x) = \dfrac{1}{\ln x}.

Example 11

hard
Restrict the domain of f(x)=cosโกxf(x) = \cos x to make it invertible on an interval containing x=2ฯ€x = 2\pi, then state the inverse's range on that branch.

Example 12

medium
Find the inverse of f(x)=x2f(x) = x^2 restricted to xโ‰ฅ0x \ge 0.

Example 13

medium
For f(x)=xโˆ’3f(x) = \sqrt{x-3}, find the natural domain restriction.

Example 14

medium
Restrict the domain of f(x)=x2โˆ’4x+7f(x) = x^2 - 4x + 7 so that the function has an inverse with range containing all xโ‰ฅ3x \ge 3.

Example 15

medium
Find the natural domain of f(x)=lnโก(x+2)xโˆ’1f(x) = \dfrac{\ln(x + 2)}{x - 1}.

Example 16

easy
Find the natural domain of f(x)=5โˆ’xf(x) = \sqrt{5 - x}.

Example 17

medium
Restrict f(x)=tanโกxf(x) = \tan x to a one-to-one interval for its inverse.

Example 18

easy
On what restricted domain is f(x)=x2f(x) = x^2 one-to-one and increasing?

Example 19

medium
Find the domain of g(x)=x+4x2โˆ’9g(x) = \frac{\sqrt{x+4}}{x^2 - 9}.

Example 20

easy
After restricting f(x)=x2f(x) = x^2 to xโ‰ฅ0x \ge 0, what is the range?
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