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Algebra 1 : Functions and Lines

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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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Example Question #23 : How To Find Inverse Variation

Find the inverse of: y=10x-1

Possible Answers:

$y=-\frac{1}{10}x -\frac{1}{10}$

$y=\frac{1}{10}x + \frac{1}{10}$

$y=\frac{1}{10}x +1$

$y=-\frac{1}{10}x + \frac{1}{10}$

$y=\frac{1}{10}x -1$

Correct answer:

$y=\frac{1}{10}x + \frac{1}{10}$

Explanation:

Interchange the x and y variables.

x=10y-1

Solve for y.

Add one on both sides.

x+1=10y-1+1

Simplify both sides.

x+1=10y

Divide by ten on both sides.

$\frac{x+1}{10}=\frac{10y}{10}$

Simplify both sides.

The answer is: $y=\frac{1}{10}x + \frac{1}{10}$

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Example Question #21 : How To Find Inverse Variation

Find the inverse of: y=-9x-14

Possible Answers:

$y=- \frac{14}{9}x-\frac{1}{9}$

$y= \frac{1}{9}x-\frac{14}{9}$

$y=\frac{14}{9}x+\frac{1}{9}$

$y=- \frac{14}{9}x-\frac{14}{9}$

$y=- \frac{1}{9}x-\frac{14}{9}$

Correct answer:

$y=- \frac{1}{9}x-\frac{14}{9}$

Explanation:

Interchange the x and y variables.

x=-9y-14

Solve for y.

Add 14 on both sides.

x+14=-9y-14+14

Simplify both sides.

x+14=-9y

Divide by negative nine on both sides.

$\frac{x+14}{-9}=\frac{-9y}{-9}$

Simplify both sides and separate the terms.

The answer is: $y=- \frac{1}{9}x-\frac{14}{9}$

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Example Question #25 : How To Find Inverse Variation

Find the inverse of: y=2x-18

Possible Answers:

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

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

$y= \frac{1}{2}x -18$

$y= \frac{1}{2}x +18$

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

Correct answer:

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

Explanation:

Interchange the x and y variables.

x=2y-18

Solve for y. Add 18 on both sides.

x+18=2y-18+18

Simplify both sides.

x+18=2y

Divide by 2 on both sides.

$\frac{x+18}{2}=\frac{2y}{2}$

Simplify both fractions.

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

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Algebra 1 : Functions and Lines

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

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

Example Question #23 : How To Find Inverse Variation

Example Question #21 : How To Find Inverse Variation

Example Question #25 : How To Find Inverse Variation

All Algebra 1 Resources

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