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Algebra 1 : Algebraic Functions

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Example Question #9 : 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 #13 : How To Find Inverse Variation

The amount of lonliness you feel varies inversely with the number of friends you have. If having 4 friends gives you a 10 on the lonliness scale, how much lonliness do you feel if you have 100 friends?

Possible Answers:

$\small 4$

$\small 0.4$

$\small 2.5$

$\small 25$

Correct answer:

$\small 0.4$

Explanation:

If lonliness and friends are inversely proportional, we can set up an equation to solve for some missing constant, k. To make things easier to write, let's use the variable L for the lonliness scale, and the variable F for how many friends you have. First we can just set up the equation:

$\small L = \frac{k}{F}$ . We know that having 4 friends gives you a 10 on the lonliness scale, so we can plug those values in to start solving for k:

$\small 10 = \frac{k}{4}$ to solve, multiply both sides by 4:

$\small 40 = k$ Now that we know the constant is 40, we can figure out how much lonliness, L, corresponds to having 100 friends by setting up a new formula. We are still generally using $\small L = \frac{k}{F}$ , and we know k is 40 and F is 100:

$\small L = \frac{40}{100}$

We can divide to get $\small L = 0.4$

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Example Question #14 : How To Find Inverse Variation

The number of lights on in your room varies inversely with the number of monsters you think are under the bed. If you only have 1 light on in your room, you're pretty sure that there are 15 monsters under the bed. How many monsters do you suspect if you turn on 4 more lights [for a total of 5]?

Possible Answers:

$\small 1$

$\small 3$

$\small 15$

Correct answer:

$\small 3$

Explanation:

If monsters and lights vary inversely, we can set up an equation to solve for some unknown constant k. To make things easier, we can use the variable M for monsters and L for the number of lights on:

$M = \frac{k}{L}$

In the first example, there are 15 monsters under the bed and 1 light on:

$15 = \frac{k}{1}$ It's pretty easy to see that our constant, k, is 15. To solve for M when there are 5 lights on, we can go back to our original $M = \frac{k}{L}$ and plug in 15 for k and 5 for L:

$\small M = \frac{15}{5 }$ Dividing gives us an answer of 3.

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Example Question #15 : How To Find Inverse Variation

The number of math problems Ricky solves in half and hour varies inversely with the number of distractions present. Even though there were 12 distractions in his room, he was able to complete 10 problems in 30 minutes. If he needs to do 15 problems in the next half hour, how many distractions does he need to remove?

Possible Answers:

$\small 4$

$\small 120$

$\small 18$

Correct answer:

$\small 4$

Explanation:

If the number of math problems varies inversely with the number of distractions, we can write an equation to solve for some constant, k. To make things easier, let's use the variable P for the number of math problems Ricky can solve, and D for the number of distractions. We can write this in a couple ways, but it will be easier later if we write it as:

$\small D = \frac{k}{P}$ . We could just as easily have written it as $\small P = \frac{k}{D}$ , and actually those equations are equivalent, or the same.

We know that Ricky can solve 10 problems with 12 distractions, so we can plug in 10=P and 12=D to solve for k:

$\small 12 = \frac{k}{10}$ to solve for k, multiply both sides by 10:

$\small 120 = k$ Now that we know k, we can figure out how many distractions Ricky can handle in order to solve 15 problems in the same amount of time. We'll do this by pluggin in 120 for k, and 15 for P:

$\small D = \frac{120}{15 }$

Dividing gives us an answer of $\small D=8$ . This means that he can only handle 8 distractions. The question is asking us how many he needs to remove. Right now he has 12, so to get only 8 he'd have to remove 4.

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Example Question #931 : Functions And Lines

Find the inverse of the following algebraic function:

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

Possible Answers:

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

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

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

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

y=x+3

Correct answer:

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

Explanation:

To find the inverse, switch the placement of the and variables:

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

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

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

Next, should be isolated, providing the inverse function:

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

xy-3x=1

xy=1+3x

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

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

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

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Example Question #17 : How To Find Inverse Variation

Find the inverse of: f(x)=3x+6

Possible Answers:

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

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

f(x)=-3x+2

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

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

Correct answer:

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

Explanation:

The notation f(x) means a function in terms of . Replace this notation with .

y=3x+6

Interchange the and variables.

x=3y+6

Solve for . Subtract six on both sides.

x-6=3y+6-6

x-6=3y

Divide by three on both sides to isolate .

$y=\frac{x-6}{3} = \frac{1}{3}x-2$

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

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Example Question #18 : How To Find Inverse Variation

If f(x) = 4x+7 , what is $f^{-1}(x)$ ?

Possible Answers:

y= -4x-7

$y= \frac{1}{4}x+\frac{7}{4}$

$y= \frac{1}{4}x-\frac{7}{4}$

$y=- \frac{1}{4}x-\frac{7}{4}$

y= 4x-7

Correct answer:

$y= \frac{1}{4}x-\frac{7}{4}$

Explanation:

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

First, replace f(x) with .

y = 4x+7

Swap the variables.

x = 4y+7

Solve for . Subtract seven on both sides.

x -7= 4y

Divide by four on both sides.

$\frac{x -7}{4}= \frac{4y}{4}$

Simplify both sides.

The inverse function is: $y= \frac{1}{4}x-\frac{7}{4}$

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Example Question #19 : How To Find Inverse Variation

Given f(x)=2x-9 , find $f^{-1}(x)$ .

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To determine the inverse function, first replace the f(x) with .

y=2x-9

Interchange the x and y variables.

x=2y-9

Solve for y. Add nine on both sides.

x+9=2y-9+9

x+9=2y

Divide by two on both sides.

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

Simplify both sides.

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

The answer is: $y=\frac{1}{2}x+\frac{9}{2}$

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Example Question #21 : How To Find Inverse Variation

Find the inverse: y=4x-6

Possible Answers:

y=-4x-6

$y= \frac{1}{4}x + \frac{3}{2}$

$y= \frac{1}{4}x- \frac{3}{2}$

y=-4x+6

$y= -\frac{1}{4}x + \frac{3}{2}$

Correct answer:

$y= \frac{1}{4}x + \frac{3}{2}$

Explanation:

To find the inverse, first interchange the x and y variables in the equation.

x=4y-6

Solve for y. Add 6 on both sides.

x+6=4y-6+6

Simplify.

x+6=4y

Divide by four on both sides.

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

Simplify both sides.

$y=\frac{x+6}{4} = \frac{1}{4}x + \frac{6}{4}= \frac{1}{4}x + \frac{3}{2}$

The inverse is: $y= \frac{1}{4}x + \frac{3}{2}$

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Example Question #22 : How To Find Inverse Variation

Find the inverse of: y=2(x+3)

Possible Answers:

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

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

y=-2x-3

y=-2x+3

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

Correct answer:

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

Explanation:

Interchange the x and y variables.

x=2(y+3)

Solve for y. First distribute the two inside the parentheses.

x=2y+6

Subtract six from both sides.

x-6=2y+6-6

Simplify.

x-6=2y

Divide by two on both sides.

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

Simplify and reduce.

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

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Algebra 1 : Algebraic Functions

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #9 : Inverse Functions

Example Question #13 : How To Find Inverse Variation

Example Question #14 : How To Find Inverse Variation

Example Question #15 : How To Find Inverse Variation

Example Question #931 : Functions And Lines

Example Question #17 : How To Find Inverse Variation

Example Question #18 : How To Find Inverse Variation

Example Question #19 : How To Find Inverse Variation

Example Question #21 : How To Find Inverse Variation

Example Question #22 : How To Find Inverse Variation

All Algebra 1 Resources

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