Question 1

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.  
-Is 4 a function value of f?

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 2

Subtract. Simplify the result, if possible.
-$\frac { 15 } { 8 x ^ { 2 } } - \frac { 12 } { 8 x ^ { 2 } }$

Accepted Answer

A)  3 
B)  $\frac { 3 } { 8 x ^ { 2 } }$ 
C)  $\frac { 3 } { 8 x ^ { 4 } }$ 
D)  $\frac { 8 } { 3 x ^ { 2 } }$ 
A)  3 
B)  $\frac { 3 } { 8 x ^ { 2 } }$ 
C)  $\frac { 3 } { 8 x ^ { 4 } }$ 
D)  $\frac { 8 } { 3 x ^ { 2 } }$

Question 3

Find the domain of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 8 x + 12 }$$

Accepted Answer

A)  domain of $f : ( \infty , - 3 ) \cup ( - 3,3 ) \cup ( 3 , \infty )$ 
B)  domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$ 
C)  domain of $f : ( \infty , 2 ) \cup ( 2,6 ) \cup ( 6 , \infty )$ 
D)  domain of $f : ( \infty , - 6 ) \cup ( - 6 , - 2 ) \cup ( - 2 , \infty )$ 
A)  domain of $f : ( \infty , - 3 ) \cup ( - 3,3 ) \cup ( 3 , \infty )$ 
B)  domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$ 
C)  domain of $f : ( \infty , 2 ) \cup ( 2,6 ) \cup ( 6 , \infty )$ 
D)  domain of $f : ( \infty , - 6 ) \cup ( - 6 , - 2 ) \cup ( - 2 , \infty )$

Question 4

Multiply as indicated.
-$$\frac { x ^ { 2 } + 7 x + 12 } { x ^ { 2 } + 9 x + 20 } \cdot \frac { x ^ { 2 } + 5 x } { x ^ { 2 } - 3 x - 18 }$$

Accepted Answer

A)  $\frac { x } { x ^ { 2 } + 9 x + 20 }$ 
B)  $\frac { x } { x - 6 }$ 
C)  $\frac { x ^ { 2 } + 5 x } { x - 6 }$ 
D)  $\frac { 1 } { x - 6 }$ 
A)  $\frac { x } { x ^ { 2 } + 9 x + 20 }$ 
B)  $\frac { x } { x - 6 }$ 
C)  $\frac { x ^ { 2 } + 5 x } { x - 6 }$ 
D)  $\frac { 1 } { x - 6 }$

Question 5

Simplify the complex fraction.
-$$\frac { \frac { 8 } { a } + 1 } { \frac { 8 } { a } - 1 }$$

Accepted Answer

A)  8 
B)  $8 - a ^ { 2 }$ 
C)  $\frac { 8 + a } { 8 - a }$ 
D)  $\frac { a ^ { 2 } } { 8 - a ^ { 2 } }$ 
A)  8 
B)  $8 - a ^ { 2 }$ 
C)  $\frac { 8 + a } { 8 - a }$ 
D)  $\frac { a ^ { 2 } } { 8 - a ^ { 2 } }$

Question 6

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { 7 x } { x + 1 } + \frac { 8 } { x - 1 } - \frac { 14 } { x ^ { 2 } - 1 }$$

Accepted Answer

A)  $\frac { 7 x - 6 } { x + 1 }$ 
B)  $\frac { 7 x } { x - 1 }$ 
C)  $\frac { 7 x - 6 } { x - 1 }$ 
D)  $\frac { x + 1 } { x - 1 }$ 
A)  $\frac { 7 x - 6 } { x + 1 }$ 
B)  $\frac { 7 x } { x - 1 }$ 
C)  $\frac { 7 x - 6 } { x - 1 }$ 
D)  $\frac { x + 1 } { x - 1 }$

Question 7

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.  
-Find f(-2).

Accepted Answer

A)  1 
B)  0 
C)  -2 
D)  2 
A)  1 
B)  0 
C)  -2 
D)  2

Question 8

Find the least common denominator of the rational expressions.
-$\frac { 15 } { x ^ { 2 } + 2 x - 15 }$ and $\frac { 4 x - 20 } { - 4 x - 20 }$

Accepted Answer

A)  $- 4 ( x - 3 ) ( x - 5 )$ 
B)  $- 4 ( x - 3 ) ( x + 5 )$ 
C)  $- 4 ( x + 3 ) ( x - 5 )$ 
D)  $- 4 ( x + 3 ) ( x + 5 )$ 
A)  $- 4 ( x - 3 ) ( x - 5 )$ 
B)  $- 4 ( x - 3 ) ( x + 5 )$ 
C)  $- 4 ( x + 3 ) ( x - 5 )$ 
D)  $- 4 ( x + 3 ) ( x + 5 )$

Question 9

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { y - 2 } { y ^ { 2 } - 1 } + \frac { 2 - y } { 1 - y ^ { 2 } }$$

Accepted Answer

A)  $\frac { 4 } { ( y + 1 ) ( y - 1 ) }$ 
B)  0 
C)  $\frac { 2 y - 4 } { ( y + 1 ) ( y - 1 ) }$ 
D)  $\frac { 2 y } { ( y + 1 ) ( y - 1 ) }$ 
A)  $\frac { 4 } { ( y + 1 ) ( y - 1 ) }$ 
B)  0 
C)  $\frac { 2 y - 4 } { ( y + 1 ) ( y - 1 ) }$ 
D)  $\frac { 2 y } { ( y + 1 ) ( y - 1 ) }$

Question 10

Multiply as indicated.
-$$\frac { 6 x - 6 } { x } \cdot \frac { 9 x ^ { 2 } } { 8 x - 8 }$$

Accepted Answer

A)  $\frac { 48 x ^ { 2 } + 96 x + 48 } { 9 x ^ { 3 } }$ 
B)  $\frac { 27 x } { 4 }$ 
C)  $\frac { 54 x ^ { 3 } - 54 x ^ { 2 } } { 8 x ^ { 2 } - 8 x }$ 
D)  $\frac { 4 } { 27 x }$ 
A)  $\frac { 48 x ^ { 2 } + 96 x + 48 } { 9 x ^ { 3 } }$ 
B)  $\frac { 27 x } { 4 }$ 
C)  $\frac { 54 x ^ { 3 } - 54 x ^ { 2 } } { 8 x ^ { 2 } - 8 x }$ 
D)  $\frac { 4 } { 27 x }$

Question 11

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$$\frac { 4 x + 20 } { 5 x ^ { 2 } + 31 x + 30 }$$

Accepted Answer

A)  $\frac { 4 x + 5 } { 5 x + 31 }$ 
B)  $\frac { 4 x } { 5 x + 6 }$ 
C)  $\frac { 4 } { 5 x + 6 }$ 
D)  Cannot be simplified 
A)  $\frac { 4 x + 5 } { 5 x + 31 }$ 
B)  $\frac { 4 x } { 5 x + 6 }$ 
C)  $\frac { 4 } { 5 x + 6 }$ 
D)  Cannot be simplified

Question 12

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.  
-How does the graph indicate that f(-4) does not exist?

Accepted Answer

A)  There is a horizontal asymptote at x = -4 indicated by a dashed line. 
B)  There is a horizontal asymptote at y = -4 indicated by a dashed line. 
C)  There is a vertical asymptote at y = -4 indicated by a dashed line. 
D)  There is a vertical asymptote at x = -4 indicated by a dashed line. 
A)  There is a horizontal asymptote at x = -4 indicated by a dashed line. 
B)  There is a horizontal asymptote at y = -4 indicated by a dashed line. 
C)  There is a vertical asymptote at y = -4 indicated by a dashed line. 
D)  There is a vertical asymptote at x = -4 indicated by a dashed line.

Question 13

Simplify the complex fraction.
-$$\frac { 4 + \frac { 2 } { x } } { \frac { x } { 3 } + \frac { 1 } { 6 } }$$

Accepted Answer

A)  1 
B)  $\frac { 12 } { y }$ 
C)  $\frac { x } { 12 }$ 
D)  12 
A)  1 
B)  $\frac { 12 } { y }$ 
C)  $\frac { x } { 12 }$ 
D)  12

Question 14

Find the least common denominator of the rational expressions.
-$\frac { 5 } { y ^ { 2 } - 4 } , \frac { 2 y } { y ^ { 2 } + 4 y + 4 }$, and $\frac { 5 y } { 2 y ^ { 2 } + 5 y + 2 }$

Accepted Answer

A)  $( y - 2 ) ( y + 2 ) ( 2 y + 1 )$ 
B)  $( y - 2 ) ( y - 2 ) ( y + 2 ) ( 2 y + 1 )$ 
C)  $( y - 2 ) ( y + 2 ) ( y + 2 ) ( 2 y + 1 )$ 
D)  $( y - 2 ) ( y + 2 )$ 
A)  $( y - 2 ) ( y + 2 ) ( 2 y + 1 )$ 
B)  $( y - 2 ) ( y - 2 ) ( y + 2 ) ( 2 y + 1 )$ 
C)  $( y - 2 ) ( y + 2 ) ( y + 2 ) ( 2 y + 1 )$ 
D)  $( y - 2 ) ( y + 2 )$

Question 15

Find the function value.
-$$f ( x ) = \frac { x ^ { 3 } - 2 } { x ^ { 2 } - 2 } ; f ( 2 )$$

Accepted Answer

A)  $\frac { 3 } { 2 }$ 
B)  4 
C)  1 
D)  3 
A)  $\frac { 3 } { 2 }$ 
B)  4 
C)  1 
D)  3

Question 16

Divide as indicated.
-$$\frac { x ^ { 2 } + 16 x + 64 } { x ^ { 2 } + 17 x + 72 } \div \frac { x ^ { 2 } + 8 x } { x ^ { 2 } + 3 x - 54 }$$

Accepted Answer

A)  $\frac { x } { x ^ { 2 } + 17 x + 72 }$ 
B)  $\frac { x - 6 } { x ^ { 2 } + 9 x }$ 
C)  $\frac { x - 6 } { x }$ 
D)  $x - 6$ 
A)  $\frac { x } { x ^ { 2 } + 17 x + 72 }$ 
B)  $\frac { x - 6 } { x ^ { 2 } + 9 x }$ 
C)  $\frac { x - 6 } { x }$ 
D)  $x - 6$

Question 17

Find the least common denominator of the rational expressions.
-$\frac { 1 } { 40 x } , \frac { 1 } { 5 x ^ { 2 } }$, and $\frac { 1 } { 8 x ^ { 3 } }$

Accepted Answer

A)  $40 x ^ { 2 }$ 
B)  $40 x ^ { 5 }$ 
C)  $8 x ^ { 3 }$ 
D)  $40 x ^ { 3 }$ 
A)  $40 x ^ { 2 }$ 
B)  $40 x ^ { 5 }$ 
C)  $8 x ^ { 3 }$ 
D)  $40 x ^ { 3 }$

Question 18

Multiply as indicated.
-$$\frac { 5 y } { 10 y + 5 } \cdot \frac { 4 y + 2 } { 7 }$$

Accepted Answer

A)  $\frac { y } { 7 }$ 
B)  $\frac { 2 y } { 7 }$ 
C)  $\frac { 2 y } { 35 }$ 
D)  $\frac { 2 } { 7 }$ 
A)  $\frac { y } { 7 }$ 
B)  $\frac { 2 y } { 7 }$ 
C)  $\frac { 2 y } { 35 }$ 
D)  $\frac { 2 } { 7 }$

Question 19

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$\frac { 15 x ^ { 2 } + 24 x + 9 } { 3 x + 3 }$

Accepted Answer

A)  $\frac { 5 x + 3 } { 3 x }$ 
B)  $\frac { 5 x + 24 } { 3 x + 5 }$ 
C)  $5 x + 3$ 
D)  Cannot be simplified 
A)  $\frac { 5 x + 3 } { 3 x }$ 
B)  $\frac { 5 x + 24 } { 3 x + 5 }$ 
C)  $5 x + 3$ 
D)  Cannot be simplified

Question 20

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.  
-Find all horizontal asymptotes of the graph.

Accepted Answer

A)  x = 4, x = -4 
B)  x = -6 
C)  y = -6 
D)  y = 4, y = -4 
A)  x = 4, x = -4 
B)  x = -6 
C)  y = -6 
D)  y = 4, y = -4

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -Is 4 a function value of f?

Subtract. Simplify the result, if possible. - $\frac { 15 } { 8 x ^ { 2 } } - \frac { 12 } { 8 x ^ { 2 } }$

$Find the domain of the rational function. - f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 8 x + 12 }$ Find the domain of the rational function. - $f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 8 x + 12 }$

Multiply as indicated. - $\frac { x ^ { 2 } + 7 x + 12 } { x ^ { 2 } + 9 x + 20 } \cdot \frac { x ^ { 2 } + 5 x } { x ^ { 2 } - 3 x - 18 }$

Simplify the complex fraction. - $\frac { \frac { 8 } { a } + 1 } { \frac { 8 } { a } - 1 }$

Perform the indicated operations. Simplify the result, if possible. - $\frac { 7 x } { x + 1 } + \frac { 8 } { x - 1 } - \frac { 14 } { x ^ { 2 } - 1 }$

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -Find f(-2).

Find the least common denominator of the rational expressions. - $\frac { 15 } { x ^ { 2 } + 2 x - 15 }$ and $\frac { 4 x - 20 } { - 4 x - 20 }$

Perform the indicated operations. Simplify the result, if possible. - $\frac { y - 2 } { y ^ { 2 } - 1 } + \frac { 2 - y } { 1 - y ^ { 2 } }$

Multiply as indicated. - $\frac { 6 x - 6 } { x } \cdot \frac { 9 x ^ { 2 } } { 8 x - 8 }$

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { 4 x + 20 } { 5 x ^ { 2 } + 31 x + 30 }$

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -How does the graph indicate that f(-4) does not exist?

Simplify the complex fraction. - $\frac { 4 + \frac { 2 } { x } } { \frac { x } { 3 } + \frac { 1 } { 6 } }$

Find the least common denominator of the rational expressions. - $\frac { 5 } { y ^ { 2 } - 4 } , \frac { 2 y } { y ^ { 2 } + 4 y + 4 }$ , and $\frac { 5 y } { 2 y ^ { 2 } + 5 y + 2 }$

Find the function value. - $f ( x ) = \frac { x ^ { 3 } - 2 } { x ^ { 2 } - 2 } ; f ( 2 )$

Divide as indicated. - $\frac { x ^ { 2 } + 16 x + 64 } { x ^ { 2 } + 17 x + 72 } \div \frac { x ^ { 2 } + 8 x } { x ^ { 2 } + 3 x - 54 }$

Find the least common denominator of the rational expressions. - $\frac { 1 } { 40 x } , \frac { 1 } { 5 x ^ { 2 } }$ , and $\frac { 1 } { 8 x ^ { 3 } }$

Multiply as indicated. - $\frac { 5 y } { 10 y + 5 } \cdot \frac { 4 y + 2 } { 7 }$

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { 15 x ^ { 2 } + 24 x + 9 } { 3 x + 3 }$

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -Find all horizontal asymptotes of the graph.

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Exam 6: Rational Expressions, Functions, and Equations

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -Is 4 a function value of f?

Subtract. Simplify the result, if possible. - 158x2−128x2\frac { 15 } { 8 x ^ { 2 } } - \frac { 12 } { 8 x ^ { 2 } }8x215​−8x212​

Find the domain of the rational function. - f(x)=x2−9x2−8x+12f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 8 x + 12 }f(x)=x2−8x+12x2−9​

Multiply as indicated. - x2+7x+12x2+9x+20⋅x2+5xx2−3x−18\frac { x ^ { 2 } + 7 x + 12 } { x ^ { 2 } + 9 x + 20 } \cdot \frac { x ^ { 2 } + 5 x } { x ^ { 2 } - 3 x - 18 }x2+9x+20x2+7x+12​⋅x2−3x−18x2+5x​

Simplify the complex fraction. - 8a+18a−1\frac { \frac { 8 } { a } + 1 } { \frac { 8 } { a } - 1 }a8​−1a8​+1​

Perform the indicated operations. Simplify the result, if possible. - 7xx+1+8x−1−14x2−1\frac { 7 x } { x + 1 } + \frac { 8 } { x - 1 } - \frac { 14 } { x ^ { 2 } - 1 }x+17x​+x−18​−x2−114​