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Precalculus : Inverse Functions

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Example Questions

Example Question #13 : Find The Inverse Of A Function

Find the inverse of y = x +4 .

Possible Answers:

$y = \frac{x+4}{4}$

x = y - 4

y = x - 4

x=4-y

$y = \frac{x-4}{4}$

Correct answer:

y = x - 4

Explanation:

To find the inverse of the function, we simply need to switch the values of and and solve for .

y = x+4

Switching and , we can write the function as:

x = y+4

We now subtract to solve for :

x - 4 = y

y = x -4

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Example Question #1054 : Pre Calculus

Find the inverse of y = 3x -7 .

Possible Answers:

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

x=7(3y)

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

$y = \frac{x+7}{3}$

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

Correct answer:

$y = \frac{x+7}{3}$

Explanation:

To find the inverse of this function, we switch and in the function:

y =3x -7

x = 3y -7

We now solve for :

x + 7 = 3y

$y = \frac{x+7}{3}$

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Example Question #14 : Find The Inverse Of A Function

Find the inverse of y = x^2 - 2 .

Possible Answers:

$y = \sqrt x + 2$

$y= \sqrt{x+2}$

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

x = y^2+2

$y=x\pm\sqrt2$

Correct answer:

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

Explanation:

To find the inverse of this function we can switch the and variables and solve for .

First, switch and in the function:

y = x^2 -2

x = y^2-2

Now, solve for :

x + 2 = y^2

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

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Example Question #12 : Find The Inverse Of A Function

Find the inverse of $y = \sqrt{x+3}$ .

Possible Answers:

y = x^2 + 3

x = y^2+3

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

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

y = x^2 - 3

Correct answer:

y = x^2 - 3

Explanation:

To find the inverse of the function, we switch and in the function.

$y = \sqrt{x+3}$

$x = \sqrt{y+3}$

We can now find our answer by solving for :

x^2 = y+3

y = x^2-3

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Example Question #11 : Find The Inverse Of A Function

Find the inverse of y = x^3 +3 .

Possible Answers:

$y = \sqrt[3]{x+3}$

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

$x = y^3 + \sqrt[3]{3}$

y=3x-3

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

Correct answer:

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

Explanation:

To find the inverse of the function, we swtich and in the function.

y = x^3 +3

x = y^3 +3

Solve for :

x - 3 = y^3

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

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Example Question #1057 : Pre Calculus

Find the inverse of this function: g(x)=(2x)^5

Possible Answers:

$g^{-1}(x)=\left({\frac{1}{2}x}\right)^{0.2}$

$g^{-1}(x)=5 \sqrt x$

$g^{-1}(x)= \frac{1}{2}x^5$

$g^{-1}(x)= \frac{x^{0.2}}{2}$

$g^{-1}(x)=\sqrt[5]{2x}$

Correct answer:

$g^{-1}(x)= \frac{x^{0.2}}{2}$

Explanation:

In order to have the inverse of a function, the new function must perform the inverse opperations in the opposite order. One way to ensure that is true is to consider the case of y=g(x) , switch x and y, then solve for y.

y=(2x)^5 in this case becomes x = (2y)^5 .

Our first step in solving is to take the reciprocal power on each side.

The reciprocal of 5 is $\frac{1}{5}=0.2$ , so we'll take both sides to the power of 0.2:

$x^{0.2}= 2y$ Now divide by 2:

$\frac{x^{0.2}}{2}=y$

Note that the answer $g^{-1}(x)=\left({\frac{1}{2}x}\right)^{0.2}$ has the correct inverse opperations, it is just in the wrong order - first you divide by 2, then you take x to the power of 0.2.

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Example Question #17 : Find The Inverse Of A Function

Find the inverse of this function: $h(x)=\sqrt{\frac{1}{x}}+1$

Possible Answers:

$h^{-1}(x)=-(x-1)^2$

$h^{-1}(x)=( \frac{1}{x}-1)^2$

$h^{-1}(x)=\frac{x^2}{1}-1$

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

$h^{-1}(x)=\frac{1}{(x-1)^2}$

Correct answer:

$h^{-1}(x)=\frac{1}{(x-1)^2}$

Explanation:

In order to have the inverse of a function, the new function must perform the inverse opperations in the opposite order. One way to ensure that is true is to consider the case of y=h(x) , switch x and y, then solve for y.

In this case, $y=\sqrt{\frac{1}{x}}+1$ becomes $x=\sqrt{\frac{1}{y}}+1$ and we solve for y.

$x=\sqrt{\frac{1}{y}}+1$ subtract 1 from both sides

$x-1 = \sqrt{\frac{1}{y}}$ square both sides

$(x-1)^2 = \frac{1}{y}$ now we will take both sides to the power of -1, in other words flip each side to the reciprocal.

We can consider (x-1)^2 to be $\frac{(x-1)^2}{1}$

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

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Example Question #103 : Functions

Find the inverse of the follow function:

y=3x^2-9

Possible Answers:

$y=\pm\sqrt{\frac{x+9}{3}}$

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

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

$y=\sqrt{\frac{x-9}{3}}$

Correct answer:

$y=\pm\sqrt{\frac{x+9}{3}}$

Explanation:

To find the inverse, substitute all x's for y's and all y's for x's and then solve for y.

x=3y^2-9

x+9=3y^2

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

$y=\pm\sqrt{\frac{x+9}{3}}$

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Example Question #104 : Functions

Find the inverse function.

Find the inverse function $f^{^{-1}}(x)$ of the function $f(x)=\frac{3x-4}{2}$

Possible Answers:

$f^{-1}(x)=\frac{2x+4}{3}$

$f^{-1}(x)= \frac{2x-4}{3}$

None of these answers are correct.

$f^{-1}(x)= \frac{2x-3}{4}$

$f^{-1}(x)= \frac{3x+4}{2}$

Correct answer:

$f^{-1}(x)=\frac{2x+4}{3}$

Explanation:

To find an inverse function, switch x and y variables and solve again for y. The new function is the inverse. f(x) can be called y. To check your answer, you can insert either function into the x variable of the other, and the equations should both solve to equal x.

$f(x)= \frac{3x-4}{2}$

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

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

2x= 3y-4

2x+4=3y

$\frac{2x+4}{3}=y$

$\frac {2x+4}{3} = f^{-1}(x)$

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Example Question #105 : Functions

Find the inverse of this function:

y=x^2 +6x + 9

Possible Answers:

$y=3+\sqrt{x}$

y=9+x

y=(x-3)^2

y= (x+3)^2

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

Correct answer:

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

Explanation:

To find the inverse of a function like this, switch the x and y variables (thereby "inverting" the function) and then solve for x.

First we would set up:

x=y^2 +6y + 9

And then let's factor:

x=(y+3)^2

And take the square root of both sides:

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

Leaving us with the final answer:

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

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Precalculus : Inverse Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #13 : Find The Inverse Of A Function

Example Question #1054 : Pre Calculus

Example Question #14 : Find The Inverse Of A Function

Example Question #12 : Find The Inverse Of A Function

Example Question #11 : Find The Inverse Of A Function

Example Question #1057 : Pre Calculus

Example Question #17 : Find The Inverse Of A Function

Example Question #103 : Functions

Example Question #104 : Functions

Example Question #105 : Functions

All Precalculus Resources

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