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Algebra II : Inverse Functions

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Example Question #11 : Inverse Functions

Inverse Functions

Given the function f(x) below, find its inverse:

f(x)=3x-5

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

When finding the inverse go through the following steps:

1) Replace f(x) with y:

y=3x-5

2) Swap the x and y variables

x=3y-5

3) Solve for y:

x+5=3y add 5 to both sides

x/3+5/3=3y/3 divide everything by 3

$f^{^{-1}}(x)=\frac{1}{3}x+\frac{5}{3}$ simplify and express as an inverse using $f^{-1}(x)$

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Example Question #12 : Inverse Functions

Find the inverse of .

y=3x^2+7

Possible Answers:

$y^{-1}= \sqrt{\frac{x-7}{3}$

$y^{-1}=\pm\sqrt{\frac{x-3}{7}}$

$y^{-1}=\frac{1}{3x^2+7}$

$y^{-1}=3\sqrt{x}+7$

$y^{-1}=\pm\sqrt{\frac{x-7}{3}}$

Correct answer:

$y^{-1}=\pm\sqrt{\frac{x-7}{3}}$

Explanation:

To find the inverse of a function, swap the x and y variable and solve for y.

The new expression after the swap is x=3y^2+7

Now solve for y.

3y^2=x-7

$y^2=\frac{x-7}{3}$

$y=\pm\sqrt{\frac{x-7}{3}}$

This y actually represents the inverse of the original y.

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Example Question #13 : Inverse Functions

What is the inverse of the following function?

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

Possible Answers:

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

$f^{-1}(x)=x^{2}+4x+5$

There is no inverse for this function.

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

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

Correct answer:

$f^{-1}(x)=x^{2}+4x+5$

Explanation:

To find the inverse of a function, switch the x and y in the original function and then solve for y.

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

$y=-2+\sqrt{x-1}$

$x=-2+\sqrt{y-1}$

$x+2=\sqrt{y-1}$

$(x+2)^{2}=(\sqrt{y-1})^{2}$

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

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

$y=x^{2}+4x+5$

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Example Question #14 : Inverse Functions

Find the inverse of y=3x-8 .

Possible Answers:

$y= \frac{1}{3}x + 8$

y=-3x+8

$y= \frac{1}{3}x - \frac{8}{3}$

$y= \frac{1}{3}x + \frac{8}{3}$

$y=\frac{8}{3}x-\frac{8}{3}$

Correct answer:

$y= \frac{1}{3}x + \frac{8}{3}$

Explanation:

In order to find the inverse, interchange the and variables.

y=3x-8

x=3y-8

Solve for . Add eight on both sides.

x+8=3y-8+8

x+8=3y

Divide by three on both sides.

$\frac{x+8}{3}=\frac{3y}{3}$

Simplify the left and right side.

The answer is: $y= \frac{1}{3}x+\frac{8}{3}$

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Example Question #15 : Inverse Functions

Find the inverse of: y=2(2-x)+2

Possible Answers:

$y= -\frac{1}{2}x + 6$

$y= -\frac{1}{2}x + 3$

$y= -\frac{1}{2}x - 6$

$y= -\frac{1}{2}x - 3$

y=2x-6

Correct answer:

$y= -\frac{1}{2}x + 3$

Explanation:

Interchange the and variables in the equation.

x=2(2-y)+2

Solve for . Subtract two on both sides.

x-2=2(2-y)

Distribute the right side of the equation to eliminate the parentheses.

x-2=4-2y

Subtract four on both sides.

x-6=-2y

Divide by negative two on both sides.

$\frac{x-6}{-2}=y$

Rewrite this in slope-intercept form.

The answer is: $y= -\frac{1}{2}x + 3$

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Example Question #16 : Inverse Functions

If f(x) = 4x^2 , find $f^{-1}(x)$ .

Possible Answers:

$y=\pm \frac{4}{x^2}$

$y=\frac{x^2}{4}$

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

$y=\pm \frac{\sqrt{x}}{4x}$

$y=\pm\frac{ \sqrt x}{2}$

Correct answer:

$y=\pm\frac{ \sqrt x}{2}$

Explanation:

The notation $f^{-1}(x)$ denotes the inverse of the function f(x) .

Rewrite f(x) using as a replacement of f(x) .

y=4x^2

Interchange and and solve for .

x=4y^2

Divide by four on both sides.

$\frac{x}{4}=y^2$

Square root both sides to eliminate the squared term.

$y=\pm \sqrt{\frac{x}{4}} = \pm\frac{ \sqrt x}{2}$

The answer is: $y=\pm\frac{ \sqrt x}{2}$

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Example Question #17 : Inverse Functions

Find the inverse of y=.5x+2x-5

Possible Answers:

None of the answers.

$y=\frac{2(x+5)}{5}$

$y=\frac{-2(x+5)}{5}$

$y=\frac{5(x+2)}{2}$

$y=\frac{2(x-5)}{5}$

Correct answer:

$y=\frac{2(x+5)}{5}$

Explanation:

y=.5x+2x-5

Combine the x terms:

y=2.5x-5

Switch the variables:

x=2.5y-5

Solve for y:

x+5=2.5y

Convert y term to a fraction to see how to simplify more easily:

$x+5=\frac{5}{2}y$

$\mathbf{y=\frac{2(x+5)}{5}}$

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Example Question #18 : Inverse Functions

Find the inverse function of $6x-\frac{2}{3}y=4$ .

Possible Answers:

$y=-\frac{1}{9}x+\frac{2}{3}$

None of these

$y=\frac{1}{9}x+\frac{2}{3}$

$y=\frac{x+6}{9}$

$y=\frac{x-9}{6}$

Correct answer:

$y=\frac{x+6}{9}$

Explanation:

$6x-\frac{2}{3}y=4$

First convert to slope intercept form:

y=9x-6

To find the inverse we switch the variables and solve for y again:

x=9y-6

x+6=9y

$\mathbf{y=\frac{x+6}{9}}$

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Example Question #19 : Inverse Functions

Find the inverse function and simplify your solution:

f(x) = 3x + 21

Possible Answers:

$f^{-1}(x) = \frac{1}{3}x - 7$

$f^{-1}(x) = 3x-7$

$f^{-1}(x) = \frac{1}{3}x + 7$

$f^{-1}(x) = -3x - 21$

$f^{-1}(x) = x -21$

Correct answer:

$f^{-1}(x) = \frac{1}{3}x - 7$

Explanation:

To find the Inverse function of:

f(x) = 3x + 21

1. Replace f(x) with :

y = 3x + 21

2. Switch the variables and :

x = 3y + 21

3. Solve for :

$y = \frac{x - 21}{3}$

4. Simplify:

$y = \frac{1}{3}x - 7$

5. Replace with $f^{-1}(x)$ and the final solution is:

$f^{-1}(x) = \frac{1}{3}x - 7$

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Example Question #19 : Inverse Functions

Find the inverse function and simplify your solution:

f(x) = 3x + 21

Possible Answers:

$f^{-1}(x) = \frac{1}{3}x - 7$

$f^{-1}(x) = 3x-7$

$f^{-1}(x) = \frac{1}{3}x + 7$

$f^{-1}(x) = -3x - 21$

$f^{-1}(x) = x -21$

Correct answer:

$f^{-1}(x) = \frac{1}{3}x - 7$

Explanation:

To find the Inverse function of:

f(x) = 3x + 21

1. Replace f(x) with :

y = 3x + 21

2. Switch the variables and :

x = 3y + 21

3. Solve for :

$y = \frac{x - 21}{3}$

4. Simplify:

$y = \frac{1}{3}x - 7$

5. Replace with $f^{-1}(x)$ and the final solution is:

$f^{-1}(x) = \frac{1}{3}x - 7$

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Algebra II : Inverse Functions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Inverse Functions

Example Question #12 : Inverse Functions

Example Question #13 : Inverse Functions

Example Question #14 : Inverse Functions

Example Question #15 : Inverse Functions

Example Question #16 : Inverse Functions

Example Question #17 : Inverse Functions

Example Question #18 : Inverse Functions

Example Question #19 : Inverse Functions

Example Question #19 : Inverse Functions

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