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Precalculus : Find the Inverse of a Function

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Example Questions

Example Question #31 : Find The Inverse Of A Function

Find the inverse of the given function:

$y=\sqrt{x+3}$

Possible Answers:

y=x^2+3

y^2=x+3

$x=\pm\sqrt{y+3}$

$x=\pm\sqrt{y-3}$

y=x^2-3

Correct answer:

y=x^2-3

Explanation:

To find the inverse function, we want to switch the values for domain in range. In other words, switch out the and variables in the function:

$y=\sqrt{x+3}$

$x=\sqrt{y+3}$

x^2=y+3

y=x^2-3

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Example Question #31 : Inverse Functions

Find the inverse of the following function:

f(x)=x+3

Possible Answers:

f(x)=x+3

f(x)=x-3

$f(x)=\frac{1}{x-3}$

$f(x)=\frac{1}{x+3}$

Correct answer:

f(x)=x-3

Explanation:

To find an inverse, simple switch f(x) and x and then solve for f(x). Thus, the inverse is:

$x=f(x)+3\Rightarrow f(x)=x-3$

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Example Question #33 : Find The Inverse Of A Function

Find the inverse of the following function:

f(x) = x^3+5

Possible Answers:

$f^{-1}(x) = \sqrt[3]{(x-5)}$

$f^{-1}(x) = \frac{1}{x^3+5}$

$f^{-1}(x) = x^3 -5$

$f^{-1}(x) = \sqrt[3]{x} -5$

Correct answer:

$f^{-1}(x) = \sqrt[3]{(x-5)}$

Explanation:

The inverse of the function f(x) = x^3+5 can be found by "reversing" the operations performed on , i.e. subtracting from the final solution, and then finding the third root of that number, or, in mathematical terms, $f^{-1}(x) = \sqrt[3]{(x-5)}$

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Example Question #41 : Inverse Functions

Find the inverse function of f(x)=2x-3 .

Possible Answers:

$f^{-1}(x)=\frac{3}{x+2}$

$f^{-1}(x)=\frac{x+3}{2}$

$f^{-1}(x)=\frac{2}{x+3}$

None of the other answers.

$f^{-1}(x)=\frac{x+2}{3}$

Correct answer:

$f^{-1}(x)=\frac{x+3}{2}$

Explanation:

To find the inverse you must reverse the variables and solve for y.

Reverse the variables:

f(x)=2x-3

y=2x-3

x=2y-3

Solve for y:

x=2y-3

x+3=2y

$y=\frac{x+3}{2}$

$f^{-1}(x)=\frac{x+3}{2}$

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Example Question #31 : Find The Inverse Of A Function

Are these two function inverses? $f(x)=\sqrt{x-4}$ and g(x)=x^2+4 .

Possible Answers:

G(x) does not have an inverse.

Cannot tell

Yes

F(x) does not have an inverse.

Correct answer:

Yes

Explanation:

One can ascertain if two functions have an inverse by finding the composition of both functions in turn. Each composition should equal x if the functions are indeed inverses of each other.

$f(g(x))=\sqrt{x^2+4-4}=x$

$g(f(x))=(\sqrt{x-4})^2+4=x$

The functions are inverses of each other.

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Example Question #43 : Inverse Functions

Find the inverse of the following equation:

y = 2(x+2)+2

Possible Answers:

$y_{inv}= \frac{1}{4}(x-1)$

$y_{inv}= \frac{1}{2}(x-6)$

$y_{inv}= \frac{1}{2}(x-1)$

$y_{inv}= \frac{1}{3}(x-2)$

Correct answer:

$y_{inv}= \frac{1}{2}(x-6)$

Explanation:

To find the inverse of a function, replace the x any y positions:

Original Equation: y = 2(x+2)+2

Inversed Equation: $x = 2(y_{inv}+2)+2$

Now solve for the inversed y value.

$x = 2(y_{inv}+2)+2$

$\frac{x-2}{2} = y_{inv}+2$

$y_{inv}=\frac{x-2}{2}-2 = \frac{1}{2}(x-6)$

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Example Question #37 : Find The Inverse Of A Function

Find the inverse of the following equation:

$y = \frac{1}{2}(x^2+4)$

Possible Answers:

$y_{inv}=(2x+5)^\frac{1}{3}$

$y_{inv}=(x+5)^2$

$y_{inv}=\sqrt{2x-4}$

$y_{inv}=(2x+2)^2$

Correct answer:

$y_{inv}=\sqrt{2x-4}$

Explanation:

To find the inverse of a function, replace the x any y positions:

Original Equation: $y = \frac{1}{2}(x^2+4)$

Inversed Equation: $x = \frac{1}{2}(y_{inv}^2+4)$

Now solve for the inversed y value.

$x = \frac{1}{2}(y_{inv}^2+4)$

$2x =(y_{inv}^2+4)$

$2x-4 =y_{inv}^2$

$y_{inv}=\sqrt{2x-4}$

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Example Question #33 : Find The Inverse Of A Function

Find the inverse of the following equation:

y = 2(x^3+12)+3

Possible Answers:

$y_{inv}=(x-7)^2$

$y_{inv}=(x-8)^\frac{1}{3}$

$y_{inv}=(2x-12)^\frac{2}{3}$

$y_{inv}=(\frac{x-27}{2})^\frac{1}{3}$

Correct answer:

$y_{inv}=(\frac{x-27}{2})^\frac{1}{3}$

Explanation:

To find the inverse of a function, replace the x any y positions:

Original Equation: y = 2(x^3+12)+3

Inversed Equation: $x = 2(y_{inv}^3+12)+3$

Now solve for the inversed y value.

$x = 2(y_{inv}^3+12)+3$

$x-3 = 2(y_{inv}^3+12)$

$\frac{x-3}{2}= y_{inv}^3+12$

$\frac{x-3}{2}-12= y_{inv}^3$

$y_{inv}=(\frac{x-3}{2}-12)^\frac{1}{3} = (\frac{x-27}{2})^\frac{1}{3}$

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Example Question #122 : Functions

Determine the inverse function, given

y=x^3+2

Possible Answers:

DNE

$y^{-1}=x^3-2$

$y^{-1}=\sqrt[3]{x-2}$

$y^{-1}=(x-2)^3$

Correct answer:

$y^{-1}=\sqrt[3]{x-2}$

Explanation:

In order to find the inverse function we

switch the variables and
solve for the new variable

For the function

y=x^3+2 ...

x=y^3+2

x-2=y^3

$y=\sqrt[3]{x-2}$

Hence, the inverse function is

$y^{-1}=\sqrt[3]{x-2}$

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Precalculus : Find the Inverse of a Function

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #31 : Find The Inverse Of A Function

Example Question #31 : Inverse Functions

Example Question #33 : Find The Inverse Of A Function

Example Question #41 : Inverse Functions

Example Question #31 : Find The Inverse Of A Function

Example Question #43 : Inverse Functions

Example Question #37 : Find The Inverse Of A Function

Example Question #33 : Find The Inverse Of A Function

Example Question #122 : Functions

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