Practice: Rational Functions

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use the graph to answer the question.

1) Find the domain and range of the rational function graphed below. 1)
10 y

-10 -8 -6 -4 -2 2 4 6 8 x
-2

-4

-6

-8

-10
A) Domain: (- ∞, -4) ∪ (-4, ∞); Range: (- ∞, ∞)
B) Domain: (- ∞, -3) ∪ (-3, ∞); Range: (- ∞, -4) ∪ (-4, ∞)
C) Domain: (- ∞, ∞); Range: (- ∞, -3) ∪ (-3, ∞)
D) Domain: (- ∞, -4) ∪ (-4, ∞); Range: (- ∞, -3) ∪ (-3, ∞)

2) Find the horizontal and vertical asymptotes of the rational function graphed below. 2)
y

-6 -4 -2 2 4 6 x
-2

-4

-6

A) Horizontal: none; vertical: x = ±4 B) Horizontal: y = 2; vertical: x = ±4

C) Horizontal: y = 0; vertical: x = ±4 D) Horizontal: y = ±4; vertical: x = 2

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 3) Find the domain and range of the rational function graphed below. 3)
y
6

-6 -4 -2 2 4 6 x
-2

-4

-6

A) Domain: (-1, ∞); Range: (0, ∞) B) Domain: (-∞ , -1) ∪ (-1, ∞); Range: (-∞ , ∞)
C) Domain: (- ∞, -1) ∪ (-1, ∞); Range: (0, ∞ ) D) Domain: (-∞ , -1) ∪ (-1, ∞); Range: (1, ∞)

Determine which of the rational functions given below has the following feature(s).
4) The x-intercept is 3 4)
x-3 3x - 1 x+3 x+9
A) f(x) = B) f(x) = C) f(x) = D) f(x) =
x+9 x+9 x+9 x-3

5) The y-intercept is 4 5)
x+4 x+8 x-8 5x - 1
A) f(x) = B) f(x) = C) f(x) = D) f(x) =
x-2 x-2 x-2 x+4

6) The horizontal asymptote is y = 5 6)

x+5 5x2 - 1 5 5x - 1
A) f(x) = B) f(x) = C) f(x) = D) f(x) =
x+9 x+9 x-5 x+9

7) The vertical asymptote is x = 3 7)

x+9 3x - 1 x+9 x-3
A) f(x) = B) f(x) = C) f(x) = D) f(x) =
x+3 x+9 x-3 x+9

8) There is a "hole" in its graph at x = 2 8)

(x - 2)(x + 9) 2x - 1
A) f(x) = B) f(x) =
(x + 5)(x - 2) x+9
x+2 x+9
C) f(x) = D) f(x) =
(x + 9)(x + 2) x-2

9) x-intercepts: 3 and 4, y-intercept: 12, vertical asymptote: x = 1, horizontal asymptote: y = 1 9)

(x + 3)(x + 4) (x - 3)(x - 4)
A) f(x) = B) f(x) =
(x + 1) (x - 1)
(x - 3)(x - 4) (x + 3)(x + 4)
C) f(x) = D) f(x) =
(x - 1)2 (x + 1)2

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 10) x-intercepts: 1 and -4, y-intercepts: none, vertical asymptotes: x = 0 and x = -5, horizontal 10)
asymptote: y = 1
(x - 1)(x + 4) (x + 1)(x - 4)
A) f(x) = B) f(x) =
(x + 5)2 (x - 5)
(x + 1)(x - 4) (x - 1)(x + 4)
C) f(x) = D) f(x) =
x(x - 5) x(x + 5)

Give the equations of any vertical asymptotes.

5
11) h(x) = 11)
x-6
A) y = 6 B) none C) x = -6 D) x = 6

8x + 8
12) f(x) = 12)
3x - 5
8 3 3 5
A) x = B) x = - C) x = D) x =
3 5 5 3

x-9
13) f(x) = 13)
x2 + 6
A) x = 3, x = -3 B) none C) x = 6 D) x = -6

(x - 7)(x + 5)
14) h(x) = 14)
x2 - 4
A) x = -7, x = 5 B) none C) x = 2, x = -2 D) x = 7, x = -5

x-7
15) f(x) = 15)
x2 + 2x
A) x = 7 B) x = 0, x = -2 C) x = -2 D) x = 2

x-7
16) g(x) = 16)
(x - 4)(x + 1)
A) x = -4, x = 1 B) x = 4, x = -1 C) x = 7 D) x = -7

5x - 1
17) h(x) = 17)
2
x + 3x - 18
A) x = -3, x = 6 B) x = 3, x = -6 C) y = 5 D) y = 3, y = -6

Give the equatiosn of any horizontal asymptotes.

x2 + 8x - 3
18) g(x) = 18)
x-3
A) y = 8 B) y = - 3 C) y = 0 D) none

x+5
19) g(x) = 19)
x2 - 6
A) none B) y = 0 C) y = 6 D) y = 1

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 14x2
20) h(x) = 20)
7x2 - 3
A) y = 3 B) none C) y = 2 D) y = 3

Sketch the graph of the rational function.

x-2
21) f(x) = 21)
x+3

A) B)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

C) D)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

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 x-3
22) f(x) = 22)
x+4

A) B)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2 2 4 6 8 10 12 -10 -8 -6 -4 -2 2 4 6 8 10 12
-2 -2
-4 -4
-6 -6
-8 -8
-10 -10

C) D)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

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 x-4
23) f(x) = 23)
x+5

A) B)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2 2 4 6 8 10 12 -10 -8 -6 -4 -2 2 4 6 8 10 12
-2 -2
-4 -4
-6 -6
-8 -8
-10 -10

C) D)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

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 4x
24) f(x) = 24)
(x + 5)(x + 1)

A) B)
y y
20 20
16 16
12 12
8 8
4 4

-10 -8 -6 -4 -2 2 4 6 8 10 x -10 -8 -6 -4 -2 2 4 6 8 10 x
-4 -4
-8 -8
-12 -12
-16 -16
-20 -20

C) D)
y y
40 10
32 8
24 6
16 4
8 2

-10 -8 -6 -4 -2-8 2 4 6 8 10 x -10 -8 -6 -4 -2-2 2 4 6 8 10 x

-16 -4
-24 -6
-32 -8
-40 -10

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 x(x + 3)
25) f(x) = 25)
(x - 5)2

A) B)
y y
12 12

8 8

4 4

-12 -8 -4 4 8 12 x -12 -8 -4 4 8 12 x
-4 -4

-8 -8

-12 -12

C) D)
y y
40
12

8
20
4

-12 -8 -4 4 8 12 x -40 -20 20 40 x

-4
-20
-8

-12
-40

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 x2 - 9
26) f(x) = 26)
x-3

A) B)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

C) D)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

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 x+1
27) f(x) = 27)
x2 + x - 20

A) B)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

C) D)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

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 x2 + 6x - 7
28) f(x) = 28)
x2 - x - 20

A) B)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

C) D)

10 10
8 8
6 6
4 4
2 2

-10 -8 -6 -4 -2-2 2 4 6 8 10 12 -10 -8 -6 -4 -2-2 2 4 6 8 10 12

-4 -4
-6 -6
-8 -8
-10 -10

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Answer Key
Testname: RATIONAL FUNCTIONS

1) D
2) B
3) C
4) A
5) C
6) D
7) C
8) A
9) C
10) D
11) D
12) D
13) B
14) C
15) B
16) B
17) B
18) D
19) B
20) C
21) B
22) D
23) D
24) A
25) C
26) B
27) A
28) A

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