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Algebra II : Functions and Graphs

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Example Questions

Example Question #3 : Inverse Functions

$g^{-1}(2)=6$

g(3)=4

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

f(3)=6

$f^{-1}(9)=2$

What is $f\circ g\circ f^{-1}(6)$ ?

Possible Answers:

Correct answer:

Explanation:

The question is essentially asking this: take $f^{-1}(6)$ say that equals , then take g(a) , then whatever that equals, say , take f(b) . So, we start with $f^{-1}(6)$ ; we know that f(3)=6 , so if we flip that around we know $f^{-1}(6)=3$ . Now we have to take g(3) , but we know that is . Now we have to take f(4) , but we don't have that in our table; we do have $f^{-1}(-2)=4$ , though, and if we flip it around, we get f(4)=-2 , which is our answer.

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Example Question #4 : Inverse Functions

$g^{-1}(2)=6$

g(3)=4

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

f(3)=6

$f^{-1}(9)=2$

What is $g\circ f\circ g^{-1}(4)$ ?

Possible Answers:

Correct answer:

Explanation:

Our question is asking "What is of of inverse?" First we find the inverse of . Looking at the question, we see g(3)=4 ; if we flip that around, we get $g^{-1}(4)=3$ . Now we need to find what f(3) is; that is an easy one, as it is directly provided: f(3)=6 . Now we need to find g(6) . Again, this isn't given, but what is given is $g^{-1}(2)=6$ , so g(6)=2 , and that is our answer.

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Example Question #5 : Inverse Functions

Over which line do you flip a function when finding its inverse?

Possible Answers:

y=-x

y=x

y=0

You do not flip a function over a line when finding its inverse.

x=0

Correct answer:

y=x

Explanation:

To find the inverse of a function, you need to change all of the values to values and all the values to values. If you flip a function over the line y=x , then you are changing all the values to values and all the values to values, giving you the inverse of your function.

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Example Question #6 : Inverse Functions

Find the inverse of this function:

$f(x)=e^{x^2-3}+\pi$

Possible Answers:

$\pm\sqrt{ln(x-\pi)+3}$

$\frac{1}{e^{x^2-3}+\pi}$

$ln(x^2-\pi+3)$

$\pm\sqrt{x-\pi}+ln(3)$

$(cos(x-\pi)+3)^2$

Correct answer:

$\pm\sqrt{ln(x-\pi)+3}$

Explanation:

To find the inverse of a function, we need to switch all the inputs ( variables) for all the outputs ( f(x) variables or variables), so if we just switch all the variables to variables and all the f(x) variables to variables and solve for , then will be our inverse function.

$f(x)=e^{x^2-3}+\pi$

turns into the following once the variables are switched:

$x=e^{y^2-3}+\pi$

the first thing we do is subtract $\pi$ from each side; then, we take the natural log of each side. This gives us

$ln(x-\pi)=y^2-3$

Then we just add three to each side and take the square root of each side, making sure we have both the positive and negative roots.

$\pm\sqrt{ln(x-\pi)+3}=y$

This is the inverse function of the function with which we were provided.

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Example Question #1 : Inverse Functions

Please find the inverse of the following function.

y=4x-9

Possible Answers:

y = -4x + 9

$x = \frac{y+9}{4}$

$y = \frac{x-9}{4}$

y = 4x + 9

$y=\frac{x+9}{4}$

Correct answer:

$y=\frac{x+9}{4}$

Explanation:

In order to find the inverse function, we must swap and and then solve for .

y = 4x-9

Becomes

x=4y-9

Now we need to solve for :

$x+\underset{+9}{0}=4y-\underset{+9}{9}$

Finally, we need to divide each side by 4.

$\frac{x+9}{4}=\frac{4y}{4}$

This gives us our inverse function:

$y = \frac{x+9}{4}$

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Example Question #1 : Inverse Functions

Find the inverse of y=3x+3 .

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To create the inverse, switch x and y making the solution x=3y+3.

y must be isolated to finish the problem.

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Example Question #1 : Inverse Functions

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

Which one of the following functions represents the inverse of $f\left ( x \right )?$

A) x-5

B) $\sqrt[3]{x^{3}-5}$

C) $\sqrt[3]{x+5}$

D) $x^{3 } + 5$

E) $x^{3} - 5$

Possible Answers:

Correct answer:

Explanation:

Given $y = x^{3} - 5$

Hence $x^{3}=y+5$

Interchanging with we get:

$y^{3}=x+5$

Solving for results in $y=\sqrt[3]{x+5}$ .

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Example Question #1 : Inverse Functions

What is the inverse of $y=\frac{2}{5}x -3$ ?

Possible Answers:

$y=\frac{5}{2}x +3$

$y=-\frac{5}{2}x+\frac{15}{2}$

$y=\frac{5}{2}x+\frac{15}{2}$

$y=\frac{5}{2}x-\frac{15}{2}$

$y=\frac{5}{2}x+15$

Correct answer:

$y=\frac{5}{2}x+\frac{15}{2}$

Explanation:

Interchange the and variables and solve for .

$x=\frac{2}{5}y -3$

$x+3=\frac{2}{5}y$

5x+15=2y

$y=\frac{5}{2}x+\frac{15}{2}$

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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 #201 : Functions And Graphs

Find the inverse of .

y=3x^2+7

Possible Answers:

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

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

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

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

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

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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Algebra II : Functions and Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3 : Inverse Functions

Example Question #4 : Inverse Functions

Example Question #5 : Inverse Functions

Example Question #6 : Inverse Functions

Example Question #1 : Inverse Functions

Example Question #1 : Inverse Functions

Example Question #1 : Inverse Functions

Example Question #1 : Inverse Functions

Example Question #11 : Inverse Functions

Example Question #201 : Functions And Graphs

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