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Precalculus : Inverse Functions

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Example Questions

Example Question #21 : Find The Inverse Of A Function

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

Possible Answers:

$\frac{4}{3}$

$\frac{1}{4}$

$-\frac{3}{4}$

$\frac{1}{3}$

$\frac{2}{3}$

Correct answer:

$\frac{1}{4}$

Explanation:

Set f(x)=y , thus $y=\frac{1+3x}{2-x}$ .

Now switch with .

So now,

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

Simplify to isolate by itself.

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

Therefore,

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

Now substitute with $f^{-1}(x)$ ,

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

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

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Example Question #22 : Find The Inverse Of A Function

Find the inverse of this function: $f(x)=\frac{8}{3}x-8$

Possible Answers:

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

$f^{-1}(x)=\frac{8}{3}x+8$

$f^{-1}(x)=\frac{3}{8}x+3$

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

$f^{-1}(x)=\frac{8}{3}x+3$

Correct answer:

$f^{-1}(x)=\frac{3}{8}x+3$

Explanation:

$f(x)=\frac{8}{3}x-8$

Write the equation in terms of x and y:

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

Switch the x and y (this inverts the relationship of the two variables):

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

Solve for y:

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

$\frac{3}{8}(x+8)=\frac{8}{3}y*\frac{3}{8}$

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

Rewrite to indicate this is the inverse:

$f^{-1}(x)=\frac{3}{8}x+3$

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Example Question #365 : Gre Subject Test: Math

Find $f^{-1}(x)$ for $f(x)=\frac{2x-7}{8x+3}$

Possible Answers:

$f^{-1}(x)=\sqrt{x^{2}+6}$

$f^{-1}(x)=\frac{2x+7}{4-3x}$

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

$f^{-1}(x)=\frac{3x+7}{2-8x}$

$f^{-1}(x)=\frac{3x^{2}+4x}{8}$

Correct answer:

$f^{-1}(x)=\frac{3x+7}{2-8x}$

Explanation:

$f(x)=\frac{2x-7}{8x+3}$

To find the inverse of a function, first swap the x and y in the given function.

$x=\frac{2y-7}{8y+3}$

Solve for y in this re-written form.

x(8y+3)=2y-7

8xy+3x=2y-7

8xy-2y=-7-3x

y(8x-2)=-7-3x

$y=\frac{-7-3x}{8x-2}$

$y=\frac{-(7+3x)}{-(-8x+2)}$

$y=\frac{3x+7}{2-8x}$

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Example Question #32 : Inverse Functions

Find the inverse of: 4x-y=5

Possible Answers:

$y=\frac{4}{5}, \frac{1}{4}$

y=4x+5

$y= \frac{1}{4}x + \frac{5}{4}$

$y= \frac{1}{4}x + 5$

y=4x-5

Correct answer:

$y= \frac{1}{4}x + \frac{5}{4}$

Explanation:

Interchange the variables and solve for .

4y-x=5

Add on both sides.

4y=x+5

Divide by four on both sides.

$\frac{4y}{4}=\frac{x+5}{4}$

The answer is: $y= \frac{1}{4}x + \frac{5}{4}$

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Example Question #24 : Find The Inverse Of A Function

Find the inverse function ( $f^{-1}$ ) of the function $f(x)=\sqrt{6x-4}$

Possible Answers:

$f^{-1}=\frac{x^2+4}{6}$

$f^{-1}=\frac{x^2-4}{6}$

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

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

None of these answers are correct.

Correct answer:

$f^{-1}=\frac{x^2+4}{6}$

Explanation:

$f(x)=\sqrt{6x-4}$

f(x) can be called y. Switch x and y, and solve for y. The resulting new equation is the inverse of f(x).

$y=\sqrt{6x-4}$

$x=\sqrt{6y-4}$

$x^2=(\sqrt{6y-4})^2$

x^2=6y-4

x^2+4=6y

$\frac{x^2+4}{6}=y$

$\frac{x^2+4}{6}=f^{-1}(x)$

To double check your work, substitute f(x) into its inverse or vice versa. Both substitutions should equal x.

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Example Question #181 : Algebra

Which of the following is the inverse of h(x) ?

h(x)=3lnx

Possible Answers:

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

$y=9e^{x}$

$y=3e^{x}$

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

Correct answer:

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

Explanation:

Which of the following is the inverse of h(x) ?

h(x)=3lnx

To find the inverse of a function, we need to swap x and y, and then rearrange to solve for y. The inverse of a function is basically the function we get if we swap the x and y coordinates for every point on the original function.

So, to begin, we can replace the h(x) with y.

y=3lnx

Next, swap x and y

x=3lny

Now, we need to get y all by itself; we can to begin by dividng the three over.

$\frac{1}{3}x=lny$

Now, recall that

And that we can rewrite any log as an exponent as follows:

log_b(a)=x

b^x=a

So with that in mind, we can rearrange our function to get y by itself:

$\frac{1}{3}x=lny$

Becomes our final answer:

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

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Example Question #31 : Inverse Functions

Find the inverse function of this function: $f(x)=\sqrt{6x-5}$ .

Possible Answers:

$f^{-1}(x)=\frac{x^2-5}{6}$

$f^{-1}(x)=\frac{x^2+5}{6}$

$f^{-1}(x)=\frac{x^2+6}{5}$

$f^{-1}(x)=\frac{x^2-6}{5}$

The inverse of this function is not a function.

Correct answer:

$f^{-1}(x)=\frac{x^2+5}{6}$

Explanation:

$f(x)=\sqrt{6x-5}$

$y=\sqrt{6x-5}$

Interchange the variables:

$x=\sqrt{6y-5}$

Solve for y:

$x^2=(\sqrt{6y-5})^2$

x^2=6y-5

x^2+5=6y

$\frac{x^2+5}{6}=y$

$f^{-1}(x)=\frac{x^2+5}{6}$

Because f(x) passes the horizontal line test, its inverse must be a function.

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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:

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

f(x)=x-3

f(x)=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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Precalculus : Inverse Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #21 : Find The Inverse Of A Function

Example Question #22 : Find The Inverse Of A Function

Example Question #365 : Gre Subject Test: Math

Example Question #32 : Inverse Functions

Example Question #24 : Find The Inverse Of A Function

Example Question #181 : Algebra

Example Question #31 : Inverse Functions

Example Question #31 : Find The Inverse Of A Function

Example Question #31 : Inverse Functions

Example Question #33 : Find The Inverse Of A Function

All Precalculus Resources

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