AnswersPrecalculusRational Functions

Rational Functions — Test Answers

25 verified answers
1
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What are the values of cap A , cap b , cap c , and cap d ?

A
cap A is equal to 1 ; cap b is equal to 0 ; cap c is equal to negative 3 ; cap d is equal to 3
B
cap A is equal to 0 ; cap b is equal to 1 ; cap c is equal to negative 3 ; cap d is equal to 3
C
cap A is equal to 1 ; cap b is equal to 0 ; cap c is equal to 3 ; cap d is equal to negative 3
D
cap A is equal to 0 ; cap b is equal to 1 ; cap c is equal to 3 ; cap d is equal to negative 3
2
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The inverse variation equation shows the relationship between wavelength in meters, x , and frequency, y y is equal to the fraction with numerator 3 times 10 to the 8th power and denominator x What is the wavelength for radio waves with frequency 3 times 10 to the 9th power ?

A
3 times 10 inverse m
B
3 times 10 to the 17th power m
C
9 times 10 to the 17th power m
D
1 times 10 inverse m
3

Which graph represents the function f of x is equal to 4 over x ?

A
Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches the y-axis as x approaches 0 from the left. Its turning point is near (negative 2 comma 2). The branch on the right approaches the y-axis as x approaches 0 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (2 comma negative 2).
B
Image with description A line graph on a Cartesian coordinate plane shows a rational function with two separate branches. The x-axis and y-axis are labeled with even integer values from negative 8 to 8. One branch lies in the first quadrant, passing through (1 comma 1) and approaching both axes without touching them. The second branch lies in the third quadrant, passing through (negative 1 comma negative 1) and also approaching both axes asymptotically.
C
Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches the y-axis as x approaches 0 from the left. Its turning point is near (negative 2 comma negative 2). The branch on the right approaches the y-axis as x approaches 0 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (2 comma 2).
D
Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches the y-axis as x approaches 0 from the left. Its turning point is near (negative 1 comma 1). The branch on the right approaches the y-axis as x approaches 0 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (1 comma negative 1).
4

Marshall bought 32 ounces of mixed nuts, which are estimated to be 30 percent peanuts. Which expression can be used to find the percentage of peanut concentration of the final mix if he adds x ounces of peanuts?

A
y is equal to the fraction with numerator 0 point 3 repeating 32 plus 0 point 3 x and denominator 32 times 100
B
y is equal to the fraction with numerator 0 point 3 repeating 32 plus x and denominator 32 plus x times 100
C
y is equal to the fraction with numerator 0 point 3 repeating 32 plus x and denominator 32 times 100
D
y is equal to the fraction with numerator 0 point 3 repeating 32 plus 0 point 3 x and denominator 32 plus x times 100
5

Using cap v is equal to l w h , what is an expression for the volume of the following prism? Open image in full-screen view

A
the fraction with numerator 4 and denominator 3 d minus 12
B
the fraction with numerator 1 and denominator 3 d minus 3
C
the fraction with numerator 4 d minus 8 and denominator 3 times open paren d minus 4 close paren squared
D
the fraction with numerator 4 times open paren d minus 2 close paren and denominator 3 open paren d minus 3 close paren open paren d minus 4 close paren
6

Which expression is equivalent to the following complex fraction? the fraction with numerator negative 2 over x plus 5 over y and denominator 3 over y minus 2 over x

A
the fraction with numerator 3 x minus 2 y and denominator negative 2 y plus 5 x
B
the fraction with numerator open paren negative 2 y plus 5 x close paren times open paren 3 x minus 2 y close paren and denominator x squared y squared
C
the fraction with numerator negative 2 y plus 5 x and denominator 3 x minus 2 y
D
the fraction with numerator x squared y squared and denominator open paren negative 2 y plus 5 x close paren times open paren 3 x minus 2 y close paren
7

Identify the graph of the rational function with an x -intercept at open paren negative 4 comma 0 close paren and open paren 4 comma 0 close paren , a vertical asymptote at x is equal to 2 , and an oblique asymptote at y is equal to x plus 2

A
Image with description On a coordinate plane, the x and y axes list negative 10 to 10. The graph has two smooth curves. One curve starts in the fourth quadrant, goes through (0 comma negative 8), and (negative 2 comma negative 3), intersects the x-axis at (negative 4 comma 0), moves left, and goes through (negative 9 comma 6). Another curve starts in the fourth quadrant, goes through (negatve 8 comma negative 8), intersects the x-axis at (4 comma 0), goes up, and passes through (3 comma 7).
B
Image with description On a coordinate plane, the x and y axes list negative 10 to 10. The graph has two smooth curves. One curve starts in the third quadrant, goes through (negative 8 comma negative 8), and (negative 5 comma negative 3), intersects the x-axis at (negative 4 comma 0), moves up, and goes through (negative 3 comma 6). Another curve starts in the third quadrant, intersects the y-axis at (0 comma negative 8), goes up, and passes through (4 comma 0). It further moves up and passes through (9 comma 6).
C
Image with description A coordinate plane where both the horizontal x-axis and vertical y-axis range from negative 10 to 10 with grid lines and labels every 5 units. The graph has two smooth curves separated by a vertical gap. The left curve extends from the bottom left, intersecting the x-axis at approximately negative 4 comma 0, and rises steeply to cross the y-axis at approximately 0 comma 7 before shooting upward. The right curve shoots upward from the bottom, intersecting the x-axis at approximately 4 comma 0, and curves more gradually up and to the right into the first quadrant.
D
Image with description On a coordinate plane, the x and y axes list negative 10 to 10. The graph has two smooth curves. One curve starts in the third quadrant, goes through (negative 3 comma negative 8), intersects the x-axis at (negative 4 comma 0), moves left, and goes through (negative 8 comma 8). Another curve starts in the fourth quadrant, goes through (negative 6 comma negative 8), intersects the x-axis at (4 comma 0), goes up, and passes through (0 comma 8).
8

Mr. Knotts found the difference of the following expression. Which statement is true about Mr. Knotts’s work? the fraction with numerator x and denominator x squared minus 1 minus the fraction with numerator 1 and denominator x minus 1 Knotts’s Work Step 1: the fraction with numerator x and denominator open paren x plus 1 close paren times open paren x minus 1 close paren minus the fraction with numerator 1 and denominator x minus 1 Step 2: the fraction with numerator x and denominator open paren x plus 1 close paren times open paren x minus 1 close paren minus the fraction with numerator 1 times open paren x plus 1 close paren and denominator open paren x plus 1 close paren times open paren x minus 1 close paren Step 3: the fraction with numerator x minus x plus 1 and denominator open paren x plus 1 close paren times open paren x minus 1 close paren Step 4: the fraction with numerator 1 and denominator open paren x plus 1 close paren times open paren x minus 1 close paren

A
In step 1, he did not correctly factor the denominator.
B
In step 4, he did not correctly subtract the numerator.
C
In step 3, he did not distribute the negative sign when he subtracted.
D
In step 2, he did not use the correct common denominator.
9

The equation cap d is equal to 13 over cap v shows that the density of a particular substance equals a mass of 13 grams divided by the volume of the substance. What happens to the density as the volume approaches 0 ?

A
The density approaches 0 .
B
The density approaches 1 .
C
The density approaches infinity.
D
The density approaches 13 .
10

Which statement is true about the discontinuities of the function f of x ? f of x is equal to the fraction with numerator x plus 1 and denominator 6 x squared minus 7 x minus 3

A
There are asymptotes at x is equal to negative 3 halves and x is equal to 1 third .
B
There are holes at x is equal to 3 halves and x is equal to negative 1 third .
C
There are asymptotes at x is equal to 3 halves and x is equal to negative 1 third .
D
There are holes at x is equal to negative 3 halves and x is equal to 1 third .
11

What is the partial fraction decomposition of the fraction with numerator 7 x squared minus 6 x plus 9 and denominator 3 x open paren 4 x squared plus 9 close paren ?

A
1 over 3 x plus the fraction with numerator x plus 2 and denominator 4 x squared plus 9
B
negative 2 over 3 x plus the fraction with numerator x minus 1 and denominator 4 x squared plus 9
C
1 over 3 x plus the fraction with numerator x minus 2 and denominator 4 x squared plus 9
D
negative 2 over 3 x plus the fraction with numerator x plus 1 and denominator 4 x squared plus 9
12

What is the difference? the fraction with numerator x and denominator x squared minus 16 minus the fraction with numerator 3 and denominator x minus 4

A
the fraction with numerator 2 times open paren x plus 6 close paren and denominator open paren x plus 4 close paren times open paren x minus 4 close paren
B
the fraction with numerator negative 2 times open paren x minus 6 close paren and denominator open paren x plus 4 close paren times open paren x minus 4 close paren
C
the fraction with numerator x minus 3 and denominator open paren x plus 5 close paren times open paren x minus 4 close paren
D
the fraction with numerator negative 2 times open paren x plus 6 close paren and denominator open paren x plus 4 close paren times open paren x minus 4 close paren
13

What is the solution to the equation the fraction with numerator y and denominator y minus 4 minus the fraction with numerator 4 and denominator y plus 4 is equal to the fraction with numerator 32 and denominator y squared minus 16 ?

A
no solution
B
y is equal to negative 4 and y is equal to 4
C
all real numbers
D
y is equal to 0
14

How many extraneous solutions does the equation have? the fraction with numerator 2 m and denominator 2 m plus 3 minus the fraction with numerator 2 m and denominator 2 m minus 3 is equal to 1

A
2
B
0
C
1
D
3
15

The table shows y , the average speed of a cyclist in miles per hour, and x , the time in hours that it takes the cyclist to complete a bicycle tour. Which rational function best models the data in the table? Time, x (hours) Average Speed, y (miles per hour) 12 8 16 6 10 and 2 thirds 9 18 5 and 1 third

A
y is equal to 2 x over 3
B
y is equal to 3 over 2 x
C
y is equal to 96 over x
D
y is equal to x over 96
16

The cost of catering a dinner is 11 point 9 5 dollars per person plus 25 dollars for delivery and setup. Which statement is true about the graph of the function that represents the average cost per person?

A
The vertical asymptote x is equal to 0 represents that the cost per person approaches 0 dollars as the number of people increases.
B
The horizontal asymptote y is equal to 11 point 9 5 represents that the cost per person approaches 11 point 9 5 dollars as the number of people increases.
C
The vertical asymptote x is equal to 11 point 9 5 represents that the cost per person approaches 11 point 9 5 dollars as the number of people increases.
D
The horizontal asymptote y is equal to 0 represents that the cost per person approaches 0 dollars as the number of people increases.
17

f of x is equal to the fraction with numerator negative 2 x squared plus 14 and denominator x squared minus 49 Which statement describes the behavior of the graph of the function shown at the vertical asymptotes?

A
As x approaches negative 7 from the left, y approaches infinity .
B
As x approaches 7 from the right, y approaches negative infinity .
C
As x approaches 7 from the left, y approaches negative infinity .
D
As x approaches negative 7 from the right, y approaches negative infinity .
18

Which statement describes the vertical asymptotes of the graph of f of x is equal to the fraction with numerator x squared minus 64 and denominator 8 x minus 64 ?

A
The graph has vertical asymptotes at both x is equal to 8 and x is equal to negative 8 .
B
The graph has a vertical asymptote at x is equal to 8 only.
C
The graph has no vertical asymptote.
D
The graph has a vertical asymptote at x is equal to negative 8 only.
19

What is the correct form of the partial fraction decomposition for the fraction with numerator 6 x plus 7 and denominator open paren x plus 2 close paren squared ?

A
the fraction with numerator cap A and denominator x plus 2 plus the fraction with numerator cap b x plus cap c and denominator open paren x plus 2 close paren squared
B
the fraction with numerator cap A and denominator x plus 2 plus the fraction with numerator cap b and denominator open paren x plus 2 close paren squared
C
cap A over x plus the fraction with numerator cap b and denominator x plus 2 plus the fraction with numerator cap c and denominator open paren x plus 2 close paren squared
D
cap A over x plus the fraction with numerator cap b and denominator x plus 2 plus the fraction with numerator cap c x plus cap d and denominator open paren x plus 2 close paren squared
20

Which expression is equivalent to the fraction with numerator m minus 4 and denominator m plus 4 divided by open paren m plus 2 close paren ?

A
the fraction with numerator m plus 4 and denominator open paren m minus 4 close paren times open paren m plus 2 close paren
B
the fraction with numerator open paren m minus 4 close paren times open paren m plus 2 close paren and denominator m plus 4
C
the fraction with numerator open paren m plus 4 close paren times open paren m plus 2 close paren and denominator m minus 4
D
the fraction with numerator m minus 4 and denominator open paren m plus 4 close paren times open paren m plus 2 close paren

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