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Algebra 1 : How to find inverse variation

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Example Question #1 : How To Find Inverse Variation

Find the inverse of the following function:

$f(x) = x^{2}-3$

Possible Answers:

$x^{2} +3$

None because the given function is not one-to-one.

$x^{2} -3$

$-\sqrt{x+3}$

$\sqrt{x+3}$

Correct answer:

None because the given function is not one-to-one.

Explanation:

$f\left ( x \right ) = x^{2}-3$

which is the same as

$y = x^{2}-3$

If we solve for we get

$x^{2} = y +3$

Taking the square root of both sides gives us the following:

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

Interchanging and gives us

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

Which is not one-to-one and therefore not a function.

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Example Question #2 : How To Find Inverse Variation

Given

$f\left ( x \right ) = 2x+3$

and

$g\left ( x \right ) = \frac{x-3}{2}$ .

Find $f\left ( g\left ( x \right ) \right )$ .

Possible Answers:

$g\left ( x \right )$

$f\left ( x \right )$

Correct answer:

Explanation:

Starting with $f\left ( x \right ) = 2x+3$

Replace with $g\left ( x \right )$ .

We get the following:

$2\left ( \frac{x-3}{2} \right )+3=x-3+3$

Which is equal to .

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Example Question #3 : How To Find Inverse Variation

Given:

f(x)=2x+3

and

$g(x)=\frac{x-3}{2}$ .

Find g(f(x)) .

Possible Answers:

f(x)

g(x)

x+y

Correct answer:

Explanation:

Start with g(x) which is equal to

$\frac{x-3}{2}$

and then replace with f(x) . We get the following:

$\frac{(2x+3)-3}{2}$

which is equal to

$\frac{2x}{2}=x$

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Example Question #4 : How To Find Inverse Variation

Which of the following is not a one-to-one function?

$f\left ( x \right ) = 2x + 5$
$f\left ( x \right ) = 2^{-x}$
$f\left ( x \right ) = x^{3} + 5$
$f\left ( x \right ) = \pm \sqrt{x+3}$
$f\left ( x \right ) = x^{5}$

Possible Answers:

Correct answer:

Explanation:

Expression 4 is not even a function because for any value of , one gets two values of violating the definition of a function. If it is not a function, then it can not be an one-to-one function.

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Example Question #5 : How To Find Inverse Variation

f(x) is a one-to-one function specified in terms of a set of $\left ( x,y \right )$ coordinates:

A = $\left \{ \left ( 0,0 \right ), \left ( 1,2\right ),\left ( 2,4 \right ), \left ( 3,6 \right )\right \}$

Which one of the following represents the inverse of the function specified by set A?

B = $\left \{ \left ( 0,0 \right ), \left ( 0,1 \right ), \left ( 0,2 \right ), \left ( 0,3 \right ) \right \}$

C = $\left \{ \left ( 0,0 \right ), \left ( 2,1 \right ), \left ( 4,2 \right ), \left ( 6,3 \right ) \right \}$

D = $\left \{ \left ( 1,1 \right ), \left ( 2,1 \right ), \left ( -2,1 \right ), \left ( 3,1 \right ) \right \}$

E = $\left \{ \left ( -2,1 \right ), \left ( -2,5 \right ), \left ( -2,3 \right ), \left ( -2, -2 \right ) \right \}$

F = $\left \{ \left ( 2,0 \right ), \left ( 2,-3 \right ),\left ( 2,5 \right ), \left ( 2,-7 \right ) \right \}$

Possible Answers:

Set C

Set F

Set E

Set D

Set B

Correct answer:

Set C

Explanation:

The set A is an one-to-one function of the form

f(x) = 2x

One can find $f^{-1}\left ( x \right )$ by interchanging the and coordinates in set A resulting in set C.

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Example Question #1 : Indirect Proportionality

varies directly with , and inversely with the square root of .

If y=6 and z =64 , then x = 10 .

Find if y=9 and z=144 .

Possible Answers:

$8\tfrac{1}{3}$

$4\tfrac{4}{9}$

$6\tfrac{2}{3}$

$12\tfrac{1}{2}$

Correct answer:

Explanation:

The variation equation can be written as below. Direct variation will put in the numerator, while inverse variation will put $\sqrt{z}$ in the denominator. is the constant that defines the variation.

$x=k* \frac{y}{\sqrt{z}}$

To find constant of variation, , substitute the values from the first scenario given in the question.

$10=k* \frac{6}{\sqrt{64}}=k* \frac{6}{8}=k* \frac{3}{4}$

$k=10\div\frac{3}{4}}= \frac{40}{3}$

We can plug this value into our variation equation.

$x=\frac{40}{3}\cdot \frac{y}{\sqrt{z}}$

Now we can solve for given the values in the second scenario of the question.

$x=\frac{40}{3}* \frac{9}{\sqrt{144}}=\frac{40}{3}* \frac{9}{12}=10$

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Example Question #2 : Indirect Proportionality

varies inversely as the square root of . If x = 15 , then A = 45 . Find if x = 30 (nearest tenth, if applicable).

Possible Answers:

63.6

22.5

11.3

31.8

Correct answer:

31.8

Explanation:

The variation equation is $A = \frac{K}{\sqrt{x}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{\sqrt{15}}$

$K = 45 \sqrt{15}$

The equation is now $A = \frac{45 \sqrt{15}}{\sqrt{x}}$ .

If x = 30 , then

$A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8$

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Example Question #6 : How To Find Inverse Variation

varies inversely as the square of . If x = 15 , then A = 45 . Find if x = 30 (nearest tenth, if applicable).

Possible Answers:

63.6

11.3

31.8

22.5

Correct answer:

11.3

Explanation:

The variation equation is $A = \frac{K}{x^{2}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{15^{2}} = \frac{K}{225}$

$K = 45 \cdot225 = 10,125$

The equation is now $A = \frac{10,125}{x^{2}}$ .

If x = 30 , then

$A = \frac{10,125}{30^{2}} = \frac{10,125}{900} \approx 11.3$

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Example Question #7 : How To Find Inverse Variation

The current, in amperes, that a battery provides an electrical object is inversely proportional to the resistance, in ohms, of the object.

A battery provides 1.2 amperes of current to a flashlight whose resistance is measured at 20 ohms. How much current will the same battery supply to a flashlight whose resistance is measured at 16 ohms?

Possible Answers:

$2 \textrm{ A}$

$1.6 \textrm{ A}$

$0.96 \textrm{ A}$

$1.8 \textrm{ A}$

$1.5 \textrm{ A}$

Correct answer:

$1.5 \textrm{ A}$

Explanation:

If is the current and is the resistance, then we can write the variation equation for some constant of variation :

$I = \frac{k}{R}$

or, alternatively,

k=IR

To find , substitute I = 1.2,R=20 :

$k=1.2 \cdot 20 = 24$

The equation is $I = \frac{24}{R}$ . Now substitute R = 16 and solve for :

$I = \frac{24}{16} = 1.5 \textrm{ A}$

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Example Question #8 : How To Find Inverse Variation

If is inversely proportional to and knowing that c=3 when d=5 , determine the proportionality constant.

Possible Answers:

Correct answer:

Explanation:

The general formula for inverse proportionality for this problem is

$c=\frac{k}{d}$

Given that c=3 when d=5 , we can find by plugging them into the formula.

$3=\frac{k}{5}$

Solve for by multiplying both sides by 5

$3\cdot5=\frac{k}{5}\cdot5$

15=k

So k=15 .

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Algebra 1 : How to find inverse variation

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Find Inverse Variation

Example Question #2 : How To Find Inverse Variation

Example Question #3 : How To Find Inverse Variation

Example Question #4 : How To Find Inverse Variation

Example Question #5 : How To Find Inverse Variation

Example Question #1 : Indirect Proportionality

Example Question #2 : Indirect Proportionality

Example Question #6 : How To Find Inverse Variation

Example Question #7 : How To Find Inverse Variation

Example Question #8 : How To Find Inverse Variation

All Algebra 1 Resources

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