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Algebra 1 : Algebraic Functions

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Example Questions

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:

g(x)

x+y

f(x)

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 #1 : 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 #2 : 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 B

Set D

Set C

Set F

Set E

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 : How To Find Inverse Variation

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:

$8\tfrac{1}{3}$

$6\tfrac{2}{3}$

$4\tfrac{4}{9}$

$12\tfrac{1}{2}$

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 #151 : Algebraic Functions

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

Possible Answers:

31.8

22.5

11.3

63.6

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 #3 : 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:

31.8

22.5

11.3

63.6

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 #31 : Proportionalities

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:

$2 \textrm{ A}$

$0.96 \textrm{ A}$

$1.5 \textrm{ A}$

$1.6 \textrm{ A}$

$1.8 \textrm{ A}$

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 #33 : Proportionalities

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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Example Question #11 : How To Find Inverse Variation

The number of days needed to construct a house is inversely proportional to the number of people that help build the house. It took 28 days to build a house with 7 people. A second house is being built and it needs to be finished in 14 days. How many people are needed to make this happen?

Possible Answers:

Correct answer:

Explanation:

The general formula of inverse proportionality for this problem is

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

where is the number of days, is the proportionality constant, and is number of people.

Before finding the number of people needed to build the house in 14 days, we need to find . Given that the house can be built in 28 days with 7 people, we have

$28=\frac{k}{7}$

Multiply both sides by 7 to find .

$28\cdot7=\frac{k}{7}\cdot7$

196=k

So k=196 . Thus,

$d=\frac{196}{p}$

Now we can find the how many people are needed to build the house in 14 days.

$14=\frac{196}{p}$

Solve for . First, multiply by on both sides:

$14\cdot p=\frac{196}{p}\cdot p$

14p=196

Divide both sides by 14

$\frac{14p}{14}=\frac{196}{14}$

p=14

So it will take 14 people to complete the house in 14 days.

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Example Question #41 : Proportionalities

The number of days to construct a house varies inversely with the number of people constructing that house. If it takes 28 days to construct a house with 6 people helping out, how long will it take if 20 people are helping out?

Possible Answers:

$10\ days$

$8.9\ days$

$8.4\ days$

$7.5\ days$

$12.1\ days$

Correct answer:

$8.4\ days$

Explanation:

The statement, 'The number of days to construct a house varies inversely with the number of people constructing that house' has the mathematical relationship $D=\frac{k}{P}$ , where D is the number of days, P is the number of people, and k is the variation constant. Given that the house can be completed in 28 days with 6 people, the k-value is calculated.

$28=\frac{k}{6}$

k=28(6)=168

This k-value can be used to find out how many days it takes to construct a house with 20 people (P = 20).

$D=\frac{168}{P}$

$D=\frac{168}{20}$

D=8.4

So it will take 8.4 days to build a house with 20 people.

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Algebra 1 : Algebraic Functions

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #3 : How To Find Inverse Variation

Example Question #1 : How To Find Inverse Variation

Example Question #2 : How To Find Inverse Variation

Example Question #1 : How To Find Inverse Variation

Example Question #151 : Algebraic Functions

Example Question #3 : How To Find Inverse Variation

Example Question #31 : Proportionalities

Example Question #33 : Proportionalities

Example Question #11 : How To Find Inverse Variation

Example Question #41 : Proportionalities

All Algebra 1 Resources

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