Deck 16: Rational Functions

Question 1

Find the domain of the function $$f ( x ) = \frac { 9 } { x - 8 }$$ . ​

A)Domain: all real numbers x except x = 8
B)Domain: all real numbers x except x = 9
C)Domain: all real numbers x except x = -9
D)Domain: all real numbers x
E)Domain: all real numbers x except x = -8

Answer 1

Domain: all real numbers x except x = 8

Question 2

Determine the equations of the vertical and horizontal asymptotes of the graph of the function $$f ( x ) = \frac { 5 x ^ { 2 } } { x ^ { 2 } - 4 }$$ .

A)horizontal: y = -2;vertical: x = 5
B)horizontal: x = 2 and x = -2;vertical: y = -5
C)horizontal: y = 5;vertical: x = 2 and x = -2
D)horizontal: x = -5;vertical: y = 2 and y = -2
E)horizontal: y = 5;vertical: x = 2

Answer 2

horizontal: y = 5;vertical: x = 2 and x = -2

Question 3

Find the domain of the function. ​​ $$g ( s ) = \frac { - 3 s } { s - 9 }$$ ​

A)s = 9
B)all real numbers s ≠ 9,s ≠ 0
C)​s = 9,s = 0
D)all real numbers s ≠ 9
E) $$\varnothing$$

Answer 3

all real numbers s ≠ 9

Question 4

The graph of the function $$f ( x ) = \frac { 3 } { x - 2 }$$ is shown below.Determine the vertical and horizontal asymptotes of its graph.

A)horizontal: y = 0;vertical: x = 2
B)horizontal: y = 0;vertical: x = -2
C)horizontal: y = -2;vertical: x = 0
D)horizontal: y = 2;vertical: x = 0
E)horizontal: y = -2;vertical: x = 2

Question 5

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 9 } { ( x - 3 ) ^ { 3 } }$$ ​

A)Domain: all real numbers x Vertical asymptote: x = 0
Horizontal asymptote: y = 0
B)Domain: all real numbers x except x = 9 Vertical asymptote: x = 0
Horizontal asymptote: y = 0
C)Domain: all real numbers x except x = 3 Vertical asymptote: x = 0
Horizontal asymptote: y = 9
D)Domain: all real numbers x Vertical asymptote: x = 0
Horizontal asymptote: y = 9
E)Domain: all real numbers x except x = 3 Vertical asymptote: x = 3
Horizontal asymptote: y = 0

Question 6

Determine the equations of the vertical and horizontal asymptotes of the graph of the function $$f ( x ) = \frac { x } { x - 4 }$$ .

A)horizontal: y = 0 ;vertical: x = 4
B)horizontal: y = -4 ;vertical: x = 0
C)horizontal: x = 0;vertical: y = 4
D)horizontal: x = 4 ;vertical: y = -1
E)horizontal: y = 1 ;vertical: x = 4

Question 7

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 7 + x } { 7 - x }$$ ​

A)Domain: all real numbers x except x = 7 Vertical asymptote: x = 7
Horizontal asymptote: y = -1
B)Domain: all real numbers x Vertical asymptote: x = -7
Horizontal asymptote: y = 0
C)Domain: all real numbers x except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = -1
D)Domain: all real numbers x except x = 7 Vertical asymptote: x = 0
Horizontal asymptote: y = -7
E)Domain: all real numbers x except x = -7 Vertical asymptote: x = 0
Horizontal asymptote: y = 7

Question 8

The graph of the function ​​ $$f ( x ) = \frac { 3 x } { x ^ { 2 } - 4 }$$ is shown below.Determine the domain.

A)Domain: all real numbers except x = -2 and x = 2
B)Domain: all real numbers except x = 2
C)Domain: all real numbers except x = 0
D)Domain: all real numbers except x = 4

Question 9

Select the correct graph of the function $$f ( x ) = \frac { 6 } { x + 4 }$$ . ​

A)​
B)​
C)​
D)​
E)​

Question 10

Determine the domain of the function $$f ( x ) = \frac { 5 } { ( x - 2 ) ^ { 4 } }$$ .

A)Domain: all real numbers except x = -5 and -2
B)Domain: all real numbers
C)Domain: all real numbers except x = -5 and 2
D)Domain: all real numbers except x = 2
E)Domain: all real numbers except x = 5 and 2

Question 11

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 49 }$$ ​

A)Domain: all real numbers x except x = 0 Vertical asymptote: x = ±7
Horizontal asymptote: No horizontal asymptote
B)Domain: all real numbers x except x = ±7 Vertical asymptote: x = ±7
Horizontal asymptote: No horizontal asymptote
C)Domain: all real numbers x except x = 7 Vertical asymptote: x = 7
Horizontal asymptote: y = 7
D)Domain: all real numbers x Vertical asymptote: x = -7
Horizontal asymptote: y = 0
E)Domain: all real numbers x except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = -49

Question 12

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 25 } { x ^ { 2 } }$$ ​

A)Domain: all real numbers x except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = 0
B)​Domain: all real numbers x except x = 5 Vertical asymptote: x = 0
Horizontal asymptote: y = 0
C)​Domain: all real numbers x except x = 25 Vertical asymptote: x = 0
Horizontal asymptote: y = 0​
D)​Domain: all real numbers x Vertical asymptote: x = 0
Horizontal asymptote: y = 0​
E)​Domain: all real numbers x except x = ±5 Vertical asymptote: y = 0
Horizontal asymptote: x = 0​

Question 13

Select the correct graph of the function $$f ( x ) = \frac { 8 } { x - 2 }$$ . ​

A)​
B)​
C)​
D)​
E)​

Question 14

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$$ ​

A)Domain: all real numbers x Vertical asymptote: No vertical asymptote
Horizontal asymptote: y = 5
B)Domain: all real numbers except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = -8
C)Domain: all real numbers x Vertical asymptote: x = -5
Horizontal asymptote: y = 0
D)Domain: all real numbers except x = 5 Vertical asymptote: No vertical asymptote
Horizontal asymptote: No horizontal asymptote
E)Domain: all real numbers except x = 9 Vertical asymptote: x = 5
Horizontal asymptote: y = 8

Question 15

Determine the equations of the vertical and horizontal asymptotes of the graph of the function $$f ( x ) = \frac { 1 } { x - 5 }$$ .

A)horizontal: y = 1;vertical: x = 5
B)horizontal: x = 0;vertical: y = 5​
C)horizontal: y = -5;vertical: x = 0
D)horizontal: x = 5;vertical: y = -1
E)horizontal: y = 0;vertical: x = 5

Question 16

Find the domain of the function $$f ( x ) = \frac { 6 x ^ { 2 } } { x ^ { 2 } - 25 }$$ . ​

A)Domain: all real numbers x except x = ​5
B)Domain: all real numbers x except x = ​±25
C)Domain: all real numbers x except x = ±6
D)Domain: all real numbers x except x = ​-5
E)Domain: all real numbers x except x = ​±5

Question 17

Select the graph of the rational function $$f ( x ) = - \frac { x + 6 } { x + 7 }$$ . ​

A)​
B)​
C)​
D)​
E)​

Question 18

Select the correct graph of the function $$f ( x ) = \frac { x - 4 } { x - 7 }$$ . ​

A)​
B)​ ​
C)​
D)​
E)​

Question 19

Find the zeros (if any)of the rational function. ​​ $$g ( x ) = \frac { x ^ { 2 } - 25 } { x + 5 }$$ ​

A)-25
B)±5
C)-5
D)25
E)5

Question 20

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 2 x ^ { 2 } } { x + 3 }$$ ​

A)Domain: all real numbers x except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = -2
B)Domain: all real numbers x except x = -3 Vertical asymptote: x = -3
Horizontal asymptote: No horizontal asymptote
C)Domain: all real numbers x except x = 3 Vertical asymptote: x = 3
Horizontal asymptote: No horizontal asymptote
D)Domain: all real numbers x except x = 3 Vertical asymptote: x = 3
Horizontal asymptote: y = 2
E)Domain: all real numbers x Vertical asymptote: x = -3
Horizontal asymptote: y = 0

Question 21

Select the correct graph for the following rational function.​ $$f ( x ) = \frac { 11 } { x - 8 }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 22

Select the correct rational function for the following graph.​ ​

A) $$f ( x ) = \frac { 3 } { x - 4 }$$
B) $f ( x ) = \frac { 3 } { x + 4 }$
C) $$f ( x ) = \frac { 3 } { x + 3 }$$
D)​ $f ( x ) = \frac { 4 } { x - 3 }$
E)​ $f ( x ) = \frac { 4 } { x - 4 }$

Question 23

Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { x - 5 } { x ^ { 2 } - 25 }$$ .

A)horizontal: y = 0;vertical: x = -5
B)horizontal: y = -5;vertical: x = 0
C)horizontal: y = 5;vertical: x = -5
D)horizontal: y = 0;vertical: x = 5
E)horizontal: y = 0;vertical: none

Question 24

Which of the following is the graph of the given equation?​ $$f ( x ) = \frac { 2 - x } { x + 3 }$$ ​

A)
B)
C)
D)​
E)

Question 25

Find the zeros (if any)of the rational function.​ $$f ( x ) = 1 + \frac { 6 } { x ^ { 2 } - 7 }$$ ​

A)±7
B)±6
C)6
D)±1
E)​7

Question 26

Find the zeros of the rational function.​ $$g ( x ) = \frac { x ^ { 2 } - 9 } { x + 3 }$$ ​

A)x = 0
B)x = -9
C)x = 3
D)x = 9
E)x = -3
​

Question 27

Determine the zeros (if any)of the rational function $$f ( x ) = \frac { x ^ { 2 } - 4 } { x - 4 }$$ .

A)x = -4,x = 4
B)no zeros
C)x = -3
D) $$x = - \frac { 2 } { 3 } , x = \frac { 2 } { 3 }$$
E)x = -2,x = 2

Question 28

Determine the zeros (if any)of the rational function $$f ( x ) = \frac { x ^ { 2 } - 4 } { x + 3 }$$ .

A)no zeros
B) $x = - \frac { 2 } { 3 } , x = \frac { 2 } { 3 }$
C)​x = -3
D)​​x = -4,​x = 4
E)​​x = -2,​x = 2

Question 29

Find domain of the following function. $$g ( x ) = \frac { x ^ { 2 } - 49 } { x ^ { 2 } - 6 x - 7 }$$ ​

A)all real numbers x except x = -49 and x = -1
B)all real numbers x except x = -6 and x = ​7
C)all real numbers x except x = 6 and x = 49
D)all real numbers x except x = 7 and x = -1
E)all real numbers x except x = -7 and x = 1

Question 30

Find vertical asymptotes of the following function.​ $$g ( x ) = \frac { x - 5 } { x ^ { 2 } - 25 }$$ ​

A)x = 25
B)x = -5
C)x = -25
D)x = 5
E)x = 0

Question 31

Find the domain of $$f ( x ) = \frac { x + 2 } { x ^ { 2 } - 4 }$$ .

A)all real numbers except x = -2
B)all real numbers
C)all real numbers except x = -2 and x = 2
D)all real numbers except x = 4
E)all real numbers except x = 2

Question 32

Find the zeros (if any)of the rational function.​ $$f ( x ) = 2 + \frac { 12 } { x ^ { 2 } + 2 }$$ ​

A)-2
B)2
C)8
D)-8
E)None

Question 33

Find the zeros (if any)of the rational function.​ $$g ( x ) = \frac { x ^ { 3 } - 8 } { x ^ { 2 } + 1 }$$ ​

A)8
B)-8
C)​2
D)-1
E)-​2

Question 34

Find the zeros (if any)of the rational function.​ $$f ( x ) = 1 - \frac { 8 } { x - 8 }$$ ​

A)​8
B)-1
C)​-16
D)1
E)​16

Question 35

Find the zeros (if any)of the rational function.​ $$f ( x ) = \frac { x ^ { 3 } - 2,744 } { x ^ { 2 } + 64 }$$ ​

A)2,744
B)-2,744
C)-14
D)​14
E)-8

Question 36

Determine the zeros (if any)of the rational function $$g ( x ) = 5 + \frac { 6 } { x ^ { 2 } + 5 }$$ .

A)no zeros
B) $x = - \sqrt { 5 } , x = \sqrt { 5 }$
C)​x = -5,x = 5
D)​ $x = - \frac { 6 } { 5 } , x = \frac { 6 } { 5 }$
E)​​x = -6

Question 37

Determine the zeros (if any)of the rational function $$g ( x ) = \frac { x ^ { 3 } - 64 } { x ^ { 2 } + 6 }$$ .

A) $x = - \sqrt { 6 } , x = \sqrt { 6 } , x = 4$
B)no zeros
C)​x = -4,x = 4
D)​x = 4
E)​ $x = - \sqrt { 6 } , x = \sqrt { 6 } , x = - 4 , x = 4$

Question 38

Find the zeros of the rational function.​ $$f ( x ) = 1 - \frac { 5 } { x - 10 }$$ ​

A)x = -15
B)x = -5
C)x = 15
D)x = -10
E)x = 5

Question 39

Determine the domain of the function $$f ( x ) = \frac { x ^ { 2 } - 25 } { x ^ { 2 } + x - 20 }$$ .

A)Domain: all real numbers except x = 5 and -4
B)Domain: all real numbers except x = -5 and -4
C)Domain: all real numbers except x = -5 and 4
D)Domain: all real numbers except x = -4
E)Domain: all real numbers

Question 40

Determine the zeros (if any)of the rational function $$g ( x ) = 6 + \frac { 3 } { x ^ { 2 } + 6 }$$ .

A)x = -6,x = 6
B) $$x = - \sqrt { 6 } , x = \sqrt { 6 }$$
C) $$x = - \frac { 1 } { 2 } , x = \frac { 1 } { 2 }$$
D)x = -3
E)no zeros

Question 41

Find domain of the following function. $$g ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } - 3 x + 2 }$$ ​

A)all real numbers x except x = -4 and x = -1
B)all real numbers x except x = -3 and x = 2
C)all real numbers x except x = 3 and x = 4
D)all real numbers x except x = 2 and x = 1
E)all real numbers x except x = -2 and x = 1

Question 42

Find the domain of $$f ( x ) = \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 3 x - 40 }$$ .

A)all real numbers
B)all real numbers except x = -8
C)all real numbers except x = 5 and x = -8
D)all real numbers except x = 8 and x = -5
E)all real numbers except x = -5,x = 5 and x = -8

Question 43

Select the correct graph of the following function. $$f ( x ) = - \frac { 1 } { ( x - 3 ) ^ { 2 } }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 44

Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 4 }$$ .

A)horizontal: y = 2;vertical: x = -2
B)horizontal: x = 1;vertical: none
C)horizontal: none;vertical: none
D)horizontal: y = 1;vertical: none
E)horizontal: y = -2;vertical: x = 1

Question 45

Find all intercepts of the following function.​ $$g ( s ) = \frac { 5 s } { s ^ { 2 } + 5 }$$ ​

A)intercept: (0,0)
B)x-intercept: (5,0)
C)x-intercept: (-5,0)
D)y-intercept: (0,-5)
E)y-intercept: (0,5)

Question 46

Find the domain of the following function​ $$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 36 }$$ ​

A)all real numbers x except x = 36
B)all real numbers x
C)all real numbers x except x = 0
D)all real numbers x except x = 6
E)all real numbers x except x = -1

Question 47

Find vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 1 } { x - 7 }$$ ​

A)Vertical asymptote: x = 0 Horizontal asymptote: y = 7
B)Vertical asymptote: x = -7 Horizontal asymptote: y = 1
C)Vertical asymptote: x = -7 Horizontal asymptote: y = 0
D)Vertical asymptote: x = 0 Horizontal asymptote: y = -7
E)Vertical asymptote: x = 7 Horizontal asymptote: y = 0

Question 48

Find domain of the following function.​ $$g ( x ) = \frac { 1 } { x + 7 }$$ ​

A)all real numbers x except x = -7
B)all real numbers x except x = 1
C)all real numbers x except x = 49
D)all real numbers x except x = 7
E)all real numbers x except x = -1

Question 49

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { x + 4 } { x ^ { 2 } - 16 }$$ ​

A)The domain is all real numbers x except x = ±4.There is a vertical asymptote at x = -4,and a horizontal asymptote at y = 0.
B)The domain is all real numbers x except x = ±4.There is a vertical asymptote at x = -16,and a horizontal asymptote at y = 0.
C)The domain is all real numbers x except x = ±16.There is a vertical asymptote at x = -16,and a horizontal asymptote at y = 0.
D)The domain is all real numbers x except x = ±4.There is a vertical asymptote at x = 4,and a horizontal asymptote at y = 0.
E)The domain is all real numbers x except x = ±16.There is a vertical asymptote at x = -4,and a horizontal asymptote at y = 0.

Question 50

Find all intercepts of the following function.​ $$g ( x ) = \frac { - 1 } { x + 3 }$$ ​

A)x-intercept: $$\left( - \frac { 1 } { 3 } , 0 \right)$$
B)y-intercept: $\left( 0 , - \frac { 1 } { 3 } \right)$
C)x-intercept: $\left( - \frac { 1 } { 3 } , 0 \right)$ and y-intercept: $\left( 0 , - \frac { 1 } { 3 } \right)$
D)x-intercept: $\left( \frac { 1 } { 3 } , 0 \right)$
E)y-intercept: $\left( 0 , \frac { 1 } { 3 } \right)$

Question 51

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$$ ​

A)The domain is all real numbers x except x = ±2.There is a vertical asymptote at x = -2,and a horizontal asymptote at y = 0.
B)The domain is all real numbers x except x = ±2.There is a vertical asymptote at x = -2,and a horizontal asymptote at y = $\frac { 1 } { 2 }$ .
C)The domain is all real numbers x.There is a vertical asymptote at x = -4,and a horizontal asymptote at y = 0.
D)The domain is all real numbers x except x ​= ±4.There is a vertical asymptote at x = -4,and a horizontal asymptote at y = 0.
E)The domain is all real numbers x except x ​= ±4.There is a vertical asymptote at x = -2,and no horizontal asymptote.

Question 52

Find domain of the following function.​ $$g ( x ) = \frac { 1 - 9 x } { 1 - x }$$ ​

A)all real numbers x
B)all real numbers x except x = 1
C)all real numbers x except x = -9
D)all real numbers x except x = -1
E)all real numbers x except x = 9

Question 53

Identify all intercepts of $$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 25 }$$ .

A)x-intercept: none;y-intercept: (0,6)
B)x-intercept: none;y-intercept: (0,1)
C)x-intercept: none;y-intercept: none
D)x-intercept: (0,0);y-intercept: (0,0)
E)x-intercepts: (-5,0)and (5,0);y-intercept: (0,1)

Question 54

Find vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 9 + 6 x } { 6 + x }$$ ​

A)Vertical asymptote: x = -6 Horizontal asymptote: y = 6
B)Vertical asymptote: x = 0 Horizontal asymptote: y = -6
C)Vertical asymptote: x = 6 Horizontal asymptote: y = -6
D)Vertical asymptote: x = 0 Horizontal asymptote: y = 6
E)Vertical asymptote: x = -6 Horizontal asymptote: y = 0

Question 55

Find the domain of $$f ( x ) = \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 4 x - 32 }$$ .

A)all real numbers except x = 8 and x = -4
B)all real numbers
C)all real numbers except x = 4 and x = -8
D)all real numbers except x = -4,x = 4 and x = -8
E)all real numbers except x = -8

Question 56

Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } + x - 12 }$$ .

A)horizontal: y = 4;vertical: x = 0
B)horizontal: y = 1;vertical: x = -4
C)horizontal: y = 1;vertical: x = 3 and x = -4
D)horizontal: y = -3;vertical: x = -4
E)horizontal: y = 0;vertical: none

Question 57

Determine the domain of $$f ( x ) = \frac { 6 x + 6 } { x ^ { 2 } - 6 x }$$ .

A)all real numbers except x = 0 and x = 6
B)all real numbers
C)all real numbers except x = 6
D)all real numbers except x = -6 and x = -1
E)all real numbers except x = -1,x = 0 and x = 6

Question 58

Find all intercepts of the following function.​ $$f ( t ) = \frac { 1 - 8 t } { t }$$ ​

A)t-intercept: $\left( - \frac { 1 } { 8 } , 0 \right)$
B)t-intercept: (0,8)
C)t-intercept: (8,0)
D)t-intercept: $\left( \frac { 1 } { 8 } , 0 \right)$
E)t-intercept: $\left( 0 , \frac { 1 } { 8 } \right)$

Question 59

Find the domain of $f ( x ) = \frac { x - 7 } { x ^ { 2 } - 49 }$ .

A)all real numbers except x = -7
B)all real numbers except x = 49
C)all real numbers except x = 7
D)all real numbers except x = 7 and x = -7
E)all real numbers

Question 60

Find all intercepts of the following function.​ $$g ( x ) = \frac { 1 } { x - 4 }$$ ​

A)y-intercept: $$\left( 0 , \frac { 1 } { 4 } \right)$$
B)x-intercept: $\left( - \frac { 1 } { 4 } , 0 \right)$
C)y-intercept: $\left( 0 , - \frac { 1 } { 4 } \right)$
D)x-intercept: $\left( \frac { 1 } { 4 } , 0 \right)$
E)x-intercept: $\left( - \frac { 1 } { 4 } , 0 \right)$ and y-intercept: $\left( 0 , - \frac { 1 } { 4 } \right)$

Question 61

Select the correct graph of the following function.​ $$f ( x ) = \frac { x ^ { 2 } - 16 } { x + 4 }$$ ​ ​

A)​
B)​
C)​
D)​
E)​

Question 62

Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { 2 x ^ { 2 } - x - 9 } { x ^ { 2 } + 3 x + 3 }$$ .

A)horizontal: y = 4;vertical: x = 0
B)horizontal: y = - 3;vertical: x = -4
C)horizontal: y = 2;vertical: none
D)horizontal: y = 1;vertical: x = - 3
E)horizontal: y = -2;vertical: x = 3 and x = -4

Question 63

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } + 10 x + 24 } { x ^ { 3 } + 6 x ^ { 2 } - x - 6 }$ .

A)horizontal: y = 1;vertical: x = -6
B)horizontal: y = 0;vertical: x = 1 and x = -6
C)horizontal: y = 0;vertical: x = -1 and x = 1
D)horizontal: none;vertical: none
E)horizontal: y = 0;vertical: x = -6 and x = -4

Question 64

Select the correct graph of the following function.​ $$f ( x ) = \frac { x ^ { 2 } + 6 } { x }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 65

Simplify f(x)and find any vertical asymptotes. ​​ $$f ( x ) = \frac { x ^ { 2 } - 4 } { x + 2 }$$ ​

A)f(x)= -x + 2;no vertical asymptotes
B)f(x)= x;no vertical asymptotes
C)f(x)= -x - 2;no vertical asymptotes
D)f(x)= x - 2;no vertical asymptotes
E)f(x)= x + 2;no vertical asymptotes

Question 66

Find all intercepts of the following function.​ $f ( x ) = \frac { 4 x ^ { 2 } - 6 x + 6 } { x - 4 }$ ​

A)y-intercept: $$\left( 0 , - \frac { 3 } { 2 } \right)$$
B)y-intercept: $\left( \frac { 3 } { 2 } , 0 \right)$
C)y-intercept: (0,3)
D)Intercept: (0,0)
E)x-intercept: $\left( - \frac { 3 } { 2 } , 0 \right)$

Question 67

Given $f ( x ) = \frac { x ^ { 2 } - 36 } { x + 6 } , g ( x ) = x - 6$ .Complete the table and explain how the two functions differ. $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 8 & - 7.5 & - 7 & - 6.5 & - 6 & - 5.5 & - 5 \\\hline f ( x ) & & & & & & & \\\hline g ( x ) & & & & & & & \\\hline\end{array}$$

A)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 8 & - 7.5 & - 7 & - 6.5 & - 6 & - 5.5 & - 5 \\\hline f ( x ) & - 14 & - 13.5 & \text { Undefined } & - 12.5 & - 12 & - 11.5 & - 11 \\\hline g ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & - 12 & - 11.5 & - 11 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -7.Because the factor x + 6 canceled,there is a hole in the graph of f(x)when x = -7.​
B)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 8 & - 7.5 & - 7 & - 6.5 & - 6 & - 5.5 & - 5 \\\hline f ( x ) & - 14 & \text { Undefined } & - 13 & - 12.5 & - 12 & - 11.5 & - 11 \\\hline g ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & - 12 & - 11.5 & - 11 \\\hline\end{array}$$
​The graph of f(x)= g(x)except when x = -7.5.Because the factor x + 6 canceled,there is a hole in the graph of f(x)when x = -7.5.​
C)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 8 & - 7.5 & - 7 & - 6.5 & - 6 & - 5.5 & - 5 \\\hline f ( x ) & - 14 & - 13.5 & - 13 & \text { Undefined } & - 12 & - 11.5 & - 11 \\\hline g ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & - 12 & - 11.5 & - 11 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -6.5.Because the factor x + 6 canceled,there is a hole in the graph of f(x)when x = -6.5.
D)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 8 & - 7.5 & - 7 & - 6.5 & - 6 & - 5.5 & - 5 \\\hline f ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & \text { Undefined } & - 11.5 & - 11 \\\hline g ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & - 12 & - 11.5 & - 11 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -6.Because the factor x + 6 canceled,there is a hole in the graph of f(x)when x = -6.​
E)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 8 & - 7.5 & - 7 & - 6.5 & - 6 & - 5.5 & - 5 \\\hline f ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & - 12 & \text { Undefined } & - 11 \\\hline g ( x ) & - 14 & - 13.5 & - 13 & - 12.5 & - 12 & - 11.5 & - 11 \\\hline\end{array}$$
​The graph of f(x)= g(x)except when x = -5.5.Because the factor x + 6 canceled,there is a hole in the graph f(x)when x = -5.5.​

Question 68

Select the correct graph of the following function. $$f ( x ) = \frac { x ^ { 2 } } { 6 x + 1 }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 69

Determine the domain of $$f ( x ) = \frac { x ^ { 2 } + 11 x + 30 } { x ^ { 3 } + 6 x ^ { 2 } - x - 6 }$$ .

A)all real numbers except x = -6 and x = -5
B)all real numbers except x = -1,x = 1,x = -6 and x = -5
C)all real numbers except x = 6
D)all real numbers except x = -1,x = 1 and x = -6
E)all real numbers

Question 70

Select the correct graph of the following function.​ $f ( x ) = \frac { x ^ { 2 } - 9 } { x }$ ​

A)​
B)​
C)​
D)​
E)​

Question 71

Given $f ( x ) = \frac { x ^ { 2 } - 10 x + 21 } { x + 5 }$ ,determine the equations of any slant and vertical asymptote.

A)slant: y = x + 9;vertical: x = -12
B)slant: y = x - 15;vertical: none
C)slant: y = x + 8;vertical: x = -4
D)slant: y = x - 15;vertical: x = -5
E)slant: y = x + 8;vertical: x = -8

Question 72

Given $$f ( x ) = \frac { x ^ { 2 } + 25 } { x + 5 } , g ( x ) = x - 5$$ .Determine the domains of f(x)and g(x). ​

A)Domain of f(x): all real numbers except -5 Domain of g(x): all real numbers except 5
B)Domain of f(x): all real numbers Domain of g(x): all real numbers except 5
C)Domain of f(x): all real numbers Domain of g(x): all real numbers
D)Domain of f(x): all real numbers except -5 Domain of g(x): all real numbers
E)Domain of f(x): all real numbers except 5 Domain of g(x): all real numbers

Question 73

Identify all intercepts of $f ( x ) = \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }$ .

A)x-intercept: none;y-intercept: none
B)x-intercept: $\left( - \frac { 7 } { 5 } , 0 \right)$ ;y-intercept: none
C)x-intercept: (-2,0);y-intercept: (0,-2)
D)x-intercepts: (-5,0), (-1,0), (1,0);y-intercept: (0,-2)
E)x-intercept: (0,0);y-intercept: (0,0)

Question 74

Find the domain of $$f ( x ) = \frac { 2 x ^ { 2 } - x - 25 } { x ^ { 2 } + 5 x + 5 }$$ .

A)all real numbers
B)all real numbers except x = -5
C)all real numbers except x = 25
D)all real numbers except x = -1
E)all real numbers except x = -5 and x = -2

Question 75

Given $$f ( x ) = \frac { 11 x + 8 } { 11 x ^ { 2 } + 8 x } , g ( x ) = \frac { 1 } { x }$$ .Determine the domains of f(x)and g(x). ​

A)Domain of f(x): all real numbers except 11 and $\frac { 8 } { 11 }$ Domain of g(x): all real numbers except 0
B)Domain of f(x): all real numbers except $- \frac { 8 } { 11 }$ Domain of g(x): all real numbers except 1
C)Domain of f(x): all real numbers Domain of g(x): all real numbers
D)Domain of f(x): all real numbers except $- \frac { 11 } { 8 }$ Domain of g(x): all real numbers except 0
E)Domain of f(x): all real numbers except 0 and $- \frac { 8 } { 11 }$ Domain of g(x): all real numbers except 0

Question 76

The cost C (in millions of dollars)of removing p% of the industrial and municipal pollutants discharged into a river is given by​ $C = \frac { 255 p } { 100 - p } , 0 \leq p \leq 100$ . ​
Select the correct graph of the cost function.
​

A)​
B)​
C)​
D)​
E)​

Question 77

Given $f ( x ) = \frac { x ^ { 2 } - 4 x + 3 } { x - 7 }$ ,determine the equations of any slant and vertical asymptote.

A)slant: y = x - 3;vertical: x = 3
B)slant: y = x + 3;vertical: x = 7
C)slant: none;vertical: none
D)slant: y = x - 11;vertical: none
E)slant: y = x + 10;vertical: x = -4

Question 78

Simplify f(x)and find any vertical asymptotes.​ $f ( x ) = \frac { x ^ { 2 } - 16 } { x + 4 }$ ​

A)f(x)= x ​- 4;no vertical asymptotes
B)f(x)= -x ​- 4;no vertical asymptotes
C)f(x)= x;no vertical asymptotes
D)f(x)= x​ + 4;no vertical asymptotes
E)f(x)= -x ​+ 4;no vertical asymptotes

Question 79

Given $$f ( x ) = \frac { x ^ { 2 } - 16 } { x + 4 } , g ( x ) = x - 4$$ .Complete the table and explain how the two functions differ. $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 6 & - 5.5 & - 5 & - 4.5 & - 4 & - 3.5 & - 3 \\\hline f ( x ) & & & & & & & \\\hline g ( x ) & & & & & & & \\\hline\end{array}$$
​

A)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 6 & - 5.5 & - 5 & - 4.5 & - 4 & - 3.5 & - 3 \\\hline f ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & \text { Undefined } & - 7.5 & - 7 \\\hline g ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & - 8 & - 7.5 & - 7 \\\hline\end{array}$$ The graph of f(x)= g(x)except when x = -4.Because the factor x + 4 canceled,there is a hole in the graph of f(x)when x = -4.
B)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 6 & - 5.5 & - 5 & - 4.5 & - 4 & - 3.5 & - 3 \\\hline f ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & - 8 & \text { Undefined } & - 7 \\\hline g ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & - 8 & - 7.5 & - 7 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -3.5.Because the factor x + 4 canceled,there is a hole in the graph of f(x)when x = -3.5.
C)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 6 & - 5.5 & - 5 & - 4.5 & - 4 & - 3.5 & - 3 \\\hline f ( x ) & - 10 & - 9.5 & - 9 & \text { Undefined } & - 8 & - 7.5 & - 7 \\\hline g ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & - 8 & - 7.5 & - 7 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -4.5.Because the factor x + 4 canceled,there is a hole in the graph of f(x)when x = -4.5.
D)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 6 & - 5.5 & - 5 & - 4.5 & - 4 & - 3.5 & - 3 \\\hline f ( x ) & - 10 & - 9.5 & \text { Undefined } & - 8.5 & - 8 & - 7.5 & - 7 \\\hline g ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & - 8 & - 7.5 & - 7 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -5.Because the factor x + 4 canceled,there is a hole in the graph of f(x)when x = -5.
E)​ $$\begin{array} { | c | c | c | c | c | c | c | c | } \hline x & - 6 & - 5.5 & - 5 & - 4.5 & - 4 & - 3.5 & - 3 \\\hline f ( x ) & - 10 & \text { Undefined } & - 9 & - 8.5 & - 8 & - 7.5 & - 7 \\\hline g ( x ) & - 10 & - 9.5 & - 9 & - 8.5 & - 8 & - 7.5 & - 7 \\\hline\end{array}$$
The graph of f(x)= g(x)except when x = -5.5.Because the factor x + 4 canceled,there is a hole in the graph of f(x)when x = -5.5.​

Question 80

Given $f ( x ) = \frac { x ^ { 2 } + 49 } { x + 7 } , g ( x ) = x - 7$ .Determine the domains of f(x)and g(x). ​

A)Domain of f(x): all real numbers except -7 Domain of g(x): all real numbers except 7
B)Domain of f(x): all real numbers Domain of g(x): all real numbers except 7
C)Domain of f(x): all real numbers Domain of g(x): all real numbers
D)Domain of f(x): all real numbers except -7 Domain of g(x): all real numbers
E)Domain of f(x): all real numbers except 7 Domain of g(x): all real numbers