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Precalculus : Functions

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Example Questions

Example Question #41 : Inverse Functions

Find the inverse function of f(x)=2x-3 .

Possible Answers:

$f^{-1}(x)=\frac{3}{x+2}$

$f^{-1}(x)=\frac{x+3}{2}$

$f^{-1}(x)=\frac{2}{x+3}$

None of the other answers.

$f^{-1}(x)=\frac{x+2}{3}$

Correct answer:

$f^{-1}(x)=\frac{x+3}{2}$

Explanation:

To find the inverse you must reverse the variables and solve for y.

Reverse the variables:

f(x)=2x-3

y=2x-3

x=2y-3

Solve for y:

x=2y-3

x+3=2y

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

$f^{-1}(x)=\frac{x+3}{2}$

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Example Question #42 : Inverse Functions

Are these two function inverses? $f(x)=\sqrt{x-4}$ and g(x)=x^2+4 .

Possible Answers:

Yes

F(x) does not have an inverse.

Cannot tell

G(x) does not have an inverse.

Correct answer:

Yes

Explanation:

One can ascertain if two functions have an inverse by finding the composition of both functions in turn. Each composition should equal x if the functions are indeed inverses of each other.

$f(g(x))=\sqrt{x^2+4-4}=x$

$g(f(x))=(\sqrt{x-4})^2+4=x$

The functions are inverses of each other.

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Example Question #43 : Inverse Functions

Find the inverse of the following equation:

y = 2(x+2)+2

Possible Answers:

$y_{inv}= \frac{1}{4}(x-1)$

$y_{inv}= \frac{1}{2}(x-6)$

$y_{inv}= \frac{1}{2}(x-1)$

$y_{inv}= \frac{1}{3}(x-2)$

Correct answer:

$y_{inv}= \frac{1}{2}(x-6)$

Explanation:

To find the inverse of a function, replace the x any y positions:

Original Equation: y = 2(x+2)+2

Inversed Equation: $x = 2(y_{inv}+2)+2$

Now solve for the inversed y value.

$x = 2(y_{inv}+2)+2$

$\frac{x-2}{2} = y_{inv}+2$

$y_{inv}=\frac{x-2}{2}-2 = \frac{1}{2}(x-6)$

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Example Question #37 : Find The Inverse Of A Function

Find the inverse of the following equation:

$y = \frac{1}{2}(x^2+4)$

Possible Answers:

$y_{inv}=(2x+5)^\frac{1}{3}$

$y_{inv}=(x+5)^2$

$y_{inv}=\sqrt{2x-4}$

$y_{inv}=(2x+2)^2$

Correct answer:

$y_{inv}=\sqrt{2x-4}$

Explanation:

To find the inverse of a function, replace the x any y positions:

Original Equation: $y = \frac{1}{2}(x^2+4)$

Inversed Equation: $x = \frac{1}{2}(y_{inv}^2+4)$

Now solve for the inversed y value.

$x = \frac{1}{2}(y_{inv}^2+4)$

$2x =(y_{inv}^2+4)$

$2x-4 =y_{inv}^2$

$y_{inv}=\sqrt{2x-4}$

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Example Question #38 : Find The Inverse Of A Function

Find the inverse of the following equation:

y = 2(x^3+12)+3

Possible Answers:

$y_{inv}=(\frac{x-27}{2})^\frac{1}{3}$

$y_{inv}=(x-7)^2$

$y_{inv}=(x-8)^\frac{1}{3}$

$y_{inv}=(2x-12)^\frac{2}{3}$

Correct answer:

$y_{inv}=(\frac{x-27}{2})^\frac{1}{3}$

Explanation:

To find the inverse of a function, replace the x any y positions:

Original Equation: y = 2(x^3+12)+3

Inversed Equation: $x = 2(y_{inv}^3+12)+3$

Now solve for the inversed y value.

$x = 2(y_{inv}^3+12)+3$

$x-3 = 2(y_{inv}^3+12)$

$\frac{x-3}{2}= y_{inv}^3+12$

$\frac{x-3}{2}-12= y_{inv}^3$

$y_{inv}=(\frac{x-3}{2}-12)^\frac{1}{3} = (\frac{x-27}{2})^\frac{1}{3}$

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Example Question #122 : Functions

Determine the inverse function, given

y=x^3+2

Possible Answers:

DNE

$y^{-1}=x^3-2$

$y^{-1}=(x-2)^3$

$y^{-1}=\sqrt[3]{x-2}$

Correct answer:

$y^{-1}=\sqrt[3]{x-2}$

Explanation:

In order to find the inverse function we

switch the variables and
solve for the new variable

For the function

y=x^3+2 ...

x=y^3+2

x-2=y^3

$y=\sqrt[3]{x-2}$

Hence, the inverse function is

$y^{-1}=\sqrt[3]{x-2}$

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Example Question #1 : Plot Points

The point (-3,-8) is in which quadrant?

Possible Answers:

Lies on an axis

III

Correct answer:

III

Explanation:

In order to determine in which quadrant the point lies, we must remember the order of the quadrants. The first quadrant is that where x and y are both positive, to the upper right of the origin. To move sequentially to the final quadrant, we go counterclockwise from the first quadrant, which means the second is where x is negative and y is positive, the third is where x and y are both negative, and the fourth is where x is positive and y is negative. We can see from our point (-3,-8) that x and y are both negative, which means the point lies in the third quadrant.

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Example Question #2 : Cartesian Coordinate System

A (2,3) and B (6,-5) are located on the circle, with forming its diameter. What is the area of the circle.

Possible Answers:

$17\pi$

$50\pi$

$20\pi$

$51\pi$

$32\pi$

Correct answer:

$20\pi$

Explanation:

Use the distance formula to find the length of .

$AB=\sqrt{(-5-3)^2+(6-2)^2}=\sqrt{64+16}=\sqrt{80}$ .

Since the length of is that of the diameter, the radius of the circle is $\frac{\sqrt{80}}{2}$ .

Thus, the area of the circle is

$A=\pi\cdot r^2=\pi\cdot \left(\frac{\sqrt{80}}{2} \right )^2=20\pi$ .

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Example Question #3 : Plot Points

Which of the following coordinates does NOT fit on the graph of the corresponding function?

$2x^{2}+6x+18$

Varsity practice precalc

Possible Answers:

(-1,14)

(3,54)

(-3,24)

(0,18)

(1,26)

Correct answer:

(-3,24)

Explanation:

When looking at the graph, it is clear that when x=-3 , has a value less than . If we were to plug in the value of x=-3 , our equation would come out as such:

$y=2(-3)^{2}+6(-3)+18$

y=18-18+18=18

Therefore, at x=-3 , we get a y=18 , providing the coordinate (-3,18) .

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Example Question #4 : Plot Points

Which of the following (x,y) coordinates does NOT correspond with the given function and graph?

$y=x^{2}-7x-18$

Varsity practice precalc

Possible Answers:

(3,-30)

(-3,8)

(0,-18)

(7,-18)

(-1,-10)

Correct answer:

(-3,8)

Explanation:

When looking at the graph, it is clear that when x=-3 , has a value greater than . When we plug in both and values into the function, it is clear that these values do not work for the function:

$8=(-3)^{2}-7(-3)-18$

8=9+21-18

$8 \neq12$

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Precalculus : Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #41 : Inverse Functions

Example Question #42 : Inverse Functions

Example Question #43 : Inverse Functions

Example Question #37 : Find The Inverse Of A Function

Example Question #38 : Find The Inverse Of A Function

Example Question #122 : Functions

Example Question #1 : Plot Points

Example Question #2 : Cartesian Coordinate System

Example Question #3 : Plot Points

Example Question #4 : Plot Points

All Precalculus Resources

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