Question 1

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

Accepted Answer

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

Question 2

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$$\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }$$

Accepted Answer

A)  $\frac { 5 y - 3 } { 3 y - 5 }$ 
B)  Cannot be simplified 
C)  $\frac { 5 y - 24 } { 3 y - 40 }$ 
D)  $\frac { y - 3 } { y - 5 }$ 
A)  $\frac { 5 y - 3 } { 3 y - 5 }$ 
B)  Cannot be simplified 
C)  $\frac { 5 y - 24 } { 3 y - 40 }$ 
D)  $\frac { y - 3 } { y - 5 }$ D

Question 3

Solve the problem.
-If the formula for the area of a triangle, $A = \frac { 1 } { 2 } b h$, is solved for $h$, then $h = \frac { 2 A } { b }$. Use this formula to find $h$ if area A is $\frac { 2 y + 1 } { 8 }$ square feet and base $b$ is $\frac { 3 y - 5 } { 11 }$ feet. Write $h$ in simplified form.

Accepted Answer

A)  $\frac { 4 ( 3 y - 5 ) } { 11 ( 2 y + 1 ) } \mathrm { ft }$ 
B)  $\frac { ( 2 \mathrm { y } + 1 ) ( 3 \mathrm { y } - 5 ) } { 44 } \mathrm { ft }$ 
C)  $\frac { 11 ( 2 y + 1 ) } { 8 ( 3 y - 5 ) } \mathrm { ft }$ 
D)  $\frac { 11 ( 2 y + 1 ) } { 4 ( 3 y - 5 ) } \mathrm { ft }$ 
A)  $\frac { 4 ( 3 y - 5 ) } { 11 ( 2 y + 1 ) } \mathrm { ft }$ 
B)  $\frac { ( 2 \mathrm { y } + 1 ) ( 3 \mathrm { y } - 5 ) } { 44 } \mathrm { ft }$ 
C)  $\frac { 11 ( 2 y + 1 ) } { 8 ( 3 y - 5 ) } \mathrm { ft }$ 
D)  $\frac { 11 ( 2 y + 1 ) } { 4 ( 3 y - 5 ) } \mathrm { ft }$ D

Question 4

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.  
-Find f(8).

Accepted Answer

A)  3 
B)  8 
C)  -10.8 
D)  -8 
A)  3 
B)  8 
C)  -10.8 
D)  -8

Question 5

Simplify the complex fraction.
-$$\frac { \frac { 4 } { 9 r - 1 } - 4 } { \frac { 4 } { 9 r - 1 } + 4 }$$

Accepted Answer

A)  $\frac { 9 r } { 2 - 9 r }$ 
B)  $\frac { 2 + 9 r } { 9 r }$ 
C)  $\frac { 2 - r } { r }$ 
D)  $\frac { 2 - 9 r } { 9 r }$ 
A)  $\frac { 9 r } { 2 - 9 r }$ 
B)  $\frac { 2 + 9 r } { 9 r }$ 
C)  $\frac { 2 - r } { r }$ 
D)  $\frac { 2 - 9 r } { 9 r }$

Question 6

Find the least common denominator of the rational expressions.
-$\frac { x + 5 } { x ^ { 2 } - 5 x - 6 }$ and $\frac { x + 9 } { x ^ { 2 } - 9 x + 18 }$

Accepted Answer

A)  $( x + 1 ) ( x - 6 )$ 
B)  $( x - 6 ) ( x - 3 )$ 
C)  $( x - 1 ) ( x + 6 ) ( x - 3 )$ 
D)  $( x + 1 ) ( x - 6 ) ( x - 3 )$ 
A)  $( x + 1 ) ( x - 6 )$ 
B)  $( x - 6 ) ( x - 3 )$ 
C)  $( x - 1 ) ( x + 6 ) ( x - 3 )$ 
D)  $( x + 1 ) ( x - 6 ) ( x - 3 )$

Question 7

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.  
-How can you tell that this is not the graph of a polynomial function?

Accepted Answer

A)  The graph is not continuous. 
B)  The value of f(1) is not equal to 1. 
C)  The graph is continuous. 
D)  The graph is not a parabola. 
A)  The graph is not continuous. 
B)  The value of f(1) is not equal to 1. 
C)  The graph is continuous. 
D)  The graph is not a parabola.

Question 8

Add. Simplify the result, if possible.
-$$\frac { 3 x + 5 } { x ^ { 2 } + 7 x + 10 } + \frac { - 3 - 2 x } { x ^ { 2 } + 7 x + 10 }$$

Accepted Answer

A)  $\frac { 1 } { x ^ { 2 } + 7 x + 10 }$ 
B)  $\frac { 1 } { x + 5 }$ 
C)  $\frac { x - 2 } { x ^ { 2 } + 7 x + 10 }$ 
D)  $\frac { 1 } { x + 2 }$ 
A)  $\frac { 1 } { x ^ { 2 } + 7 x + 10 }$ 
B)  $\frac { 1 } { x + 5 }$ 
C)  $\frac { x - 2 } { x ^ { 2 } + 7 x + 10 }$ 
D)  $\frac { 1 } { x + 2 }$

Question 9

Find the function value.
-$$f ( x ) = \frac { x - 4 } { 7 x + 12 } ; f ( - 2 )$$

Accepted Answer

A)  0 
B)  1 
C)  $- 3$ 
D)  3 
A)  0 
B)  1 
C)  $- 3$ 
D)  3

Question 10

Simplify the complex fraction.
-$$\frac { 6 x ^ { - 1 } - 4 y ^ { - 1 } } { - 5 x ^ { - 2 } - 4 y ^ { - 2 } }$$

Accepted Answer

A)  $\frac { - 6 x y ^ { 2 } + 4 x ^ { 2 } y } { 5 y ^ { 2 } - 4 x ^ { 2 } }$ 
B)  $\frac { 6 x y ^ { 2 } - 4 x ^ { 2 } y } { - 5 y ^ { 2 } - 4 x ^ { 2 } }$ 
C)  $\frac { - 5 x ^ { 2 } - 4 y ^ { 2 } } { 6 x - 4 y }$ 
D)  $\frac { - 5 y ^ { 2 } - 4 x ^ { 2 } } { 6 x y ^ { 2 } - 4 x ^ { 2 } y }$ 
A)  $\frac { - 6 x y ^ { 2 } + 4 x ^ { 2 } y } { 5 y ^ { 2 } - 4 x ^ { 2 } }$ 
B)  $\frac { 6 x y ^ { 2 } - 4 x ^ { 2 } y } { - 5 y ^ { 2 } - 4 x ^ { 2 } }$ 
C)  $\frac { - 5 x ^ { 2 } - 4 y ^ { 2 } } { 6 x - 4 y }$ 
D)  $\frac { - 5 y ^ { 2 } - 4 x ^ { 2 } } { 6 x y ^ { 2 } - 4 x ^ { 2 } y }$

Question 11

Add. Simplify the result, if possible.
-$$\frac { 7 } { 8 x ^ { 2 } } + \frac { 5 } { 8 x ^ { 2 } }$$

Accepted Answer

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

Question 12

$$\text { The rational function } f ( x ) = \frac { 120 x } { 100 - x } \text { models the cost, } f ( x ) \text {, in millions of dollars, to remove } x \% \text { of the trash from American }$$ highways. The graph is shown. Use the equation to solve the problem.

-What value of x must be excluded from the rational function's domain?

Accepted Answer

A)  10 
B)  100 
C)  1000 
D)  1 
A)  10 
B)  100 
C)  1000 
D)  1

Question 13

Multiply as indicated.
-$\frac { 2 x ^ { 2 } } { 4 } \cdot \frac { 8 } { x ^ { 3 } }$

Accepted Answer

A)  $\frac { 4 x ^ { 2 } } { x ^ { 3 } }$ 
B)  $\frac { x } { 4 }$ 
C)  $\frac { 16 x ^ { 2 } } { 4 x ^ { 3 } }$ 
D)  $\frac { 4 } { x }$ 
A)  $\frac { 4 x ^ { 2 } } { x ^ { 3 } }$ 
B)  $\frac { x } { 4 }$ 
C)  $\frac { 16 x ^ { 2 } } { 4 x ^ { 3 } }$ 
D)  $\frac { 4 } { x }$

Question 14

Find the domain of the rational function.
-$$f ( x ) = \frac { x - 3 } { 5 - x }$$

Accepted Answer

A)  domain of $\mathrm { f } : ( \infty , 5 ) \cup ( 5 , \infty )$ 
B)  domain of $f : ( \infty , \infty )$ 
C)  domain of $f : ( \infty , - 5 ) \cup ( - 5 , \infty )$ 
D)  domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$ 
A)  domain of $\mathrm { f } : ( \infty , 5 ) \cup ( 5 , \infty )$ 
B)  domain of $f : ( \infty , \infty )$ 
C)  domain of $f : ( \infty , - 5 ) \cup ( - 5 , \infty )$ 
D)  domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Question 15

Simplify the complex fraction.
-$$\frac { 9 + \frac { 3 } { x } } { \frac { x } { 4 } + \frac { 1 } { 12 } }$$

Accepted Answer

A)  1 
B)  $\frac { 36 } { x }$ 
C)  36 
D)  $\frac { \mathrm { x } } { 36 }$ 
A)  1 
B)  $\frac { 36 } { x }$ 
C)  36 
D)  $\frac { \mathrm { x } } { 36 }$

Question 16

Divide as indicated.
-$$\frac { x ^ { 2 } + 15 x + 54 } { x ^ { 2 } + 16 x + 63 } \div \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 17 x + 70 }$$

Accepted Answer

A)  $\frac { x + 10 } { x }$ 
B)  $\frac { x + 10 } { x ^ { 2 } + 7 x }$ 
C)  $x + 10$ 
D)  $\frac { x } { x ^ { 2 } + 16 x + 63 }$ 
A)  $\frac { x + 10 } { x }$ 
B)  $\frac { x + 10 } { x ^ { 2 } + 7 x }$ 
C)  $x + 10$ 
D)  $\frac { x } { x ^ { 2 } + 16 x + 63 }$

Question 17

Find the least common denominator of the rational expressions.
-$\frac { 1 } { - 3 x }$ and $\frac { 2 } { x ^ { 2 } + 7 x }$

Accepted Answer

A)  $- 3 x ( x + 7 )$ 
B)  $- 3 x + 7$ 
C)  $- 3 x ^ { 2 } + 2$ 
D)  $- 3 x ^ { 2 } + 7$ 
A)  $- 3 x ( x + 7 )$ 
B)  $- 3 x + 7$ 
C)  $- 3 x ^ { 2 } + 2$ 
D)  $- 3 x ^ { 2 } + 7$

Question 18

Add. Simplify the result, if possible.
-$\frac { 4 } { 13 x } + \frac { 6 } { 13 x }$

Accepted Answer

A)  $\frac { 13 x } { 10 }$
$26 x$ 
B)  1 
C)  $\frac { 10 } { 26 x }$ 
D)  $\frac { 10 } { 13 x }$ 
A)  $\frac { 13 x } { 10 }$
$26 x$ 
B)  1 
C)  $\frac { 10 } { 26 x }$ 
D)  $\frac { 10 } { 13 x }$

Question 19

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$$\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 11 x + 33 x ^ { 2 } }$$

Accepted Answer

A)  $\frac { 7 } { 11 }$ 
B)  $\frac { 7 + 21 x ^ { 3 } } { 11 x + 33 }$ 
C)  $\frac { 7 x } { 11 }$ 
D)  Cannot be simplified 
A)  $\frac { 7 } { 11 }$ 
B)  $\frac { 7 + 21 x ^ { 3 } } { 11 x + 33 }$ 
C)  $\frac { 7 x } { 11 }$ 
D)  Cannot be simplified

Question 20

Find the least common denominator of the rational expressions.
-$\frac { 3 } { x }$ and $\frac { 3 } { x + 6 }$

Accepted Answer

A)  $- 6$ 
B)  6 
C)  $x ( x + 6 )$ 
D)  $x + 6$ 
A)  $- 6$ 
B)  6 
C)  $x ( x + 6 )$ 
D)  $x + 6$

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

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }$

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -Find f(8).

Simplify the complex fraction. - $\frac { \frac { 4 } { 9 r - 1 } - 4 } { \frac { 4 } { 9 r - 1 } + 4 }$

Find the least common denominator of the rational expressions. - $\frac { x + 5 } { x ^ { 2 } - 5 x - 6 }$ and $\frac { x + 9 } { x ^ { 2 } - 9 x + 18 }$

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -How can you tell that this is not the graph of a polynomial function?

Add. Simplify the result, if possible. - $\frac { 3 x + 5 } { x ^ { 2 } + 7 x + 10 } + \frac { - 3 - 2 x } { x ^ { 2 } + 7 x + 10 }$

Find the function value. - $f ( x ) = \frac { x - 4 } { 7 x + 12 } ; f ( - 2 )$

Simplify the complex fraction. - $\frac { 6 x ^ { - 1 } - 4 y ^ { - 1 } } { - 5 x ^ { - 2 } - 4 y ^ { - 2 } }$

Add. Simplify the result, if possible. - $\frac { 7 } { 8 x ^ { 2 } } + \frac { 5 } { 8 x ^ { 2 } }$

Multiply as indicated. - $\frac { 2 x ^ { 2 } } { 4 } \cdot \frac { 8 } { x ^ { 3 } }$

$Find the domain of the rational function. - f ( x ) = \frac { x - 3 } { 5 - x }$ Find the domain of the rational function. - $f ( x ) = \frac { x - 3 } { 5 - x }$

Simplify the complex fraction. - $\frac { 9 + \frac { 3 } { x } } { \frac { x } { 4 } + \frac { 1 } { 12 } }$

Divide as indicated. - $\frac { x ^ { 2 } + 15 x + 54 } { x ^ { 2 } + 16 x + 63 } \div \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 17 x + 70 }$

Find the least common denominator of the rational expressions. - $\frac { 1 } { - 3 x }$ and $\frac { 2 } { x ^ { 2 } + 7 x }$

Add. Simplify the result, if possible. - $\frac { 4 } { 13 x } + \frac { 6 } { 13 x }$

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 11 x + 33 x ^ { 2 } }$

Find the least common denominator of the rational expressions. - $\frac { 3 } { x }$ and $\frac { 3 } { x + 6 }$

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Exam 6: Rational Expressions, Functions, and Equations

Find the function value. - f(x)=x−32x;f(−3)f ( x ) = \frac { x - 3 } { 2 x } ; f ( - 3 )f(x)=2xx−3​;f(−3)

Simplify the rational expression. If the rational expression cannot be simplified, so state. - y2+5y−24y2+3y−40\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }y2+3y−40y2+5y−24​

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -Find f(8).

Simplify the complex fraction. - 49r−1−449r−1+4\frac { \frac { 4 } { 9 r - 1 } - 4 } { \frac { 4 } { 9 r - 1 } + 4 }9r−14​+49r−14​−4​

Find the least common denominator of the rational expressions. - x+5x2−5x−6\frac { x + 5 } { x ^ { 2 } - 5 x - 6 }x2−5x−6x+5​ and x+9x2−9x+18\frac { x + 9 } { x ^ { 2 } - 9 x + 18 }x2−9x+18x+9​

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. -How can you tell that this is not the graph of a polynomial function?

Add. Simplify the result, if possible. - 3x+5x2+7x+10+−3−2xx2+7x+10\frac { 3 x + 5 } { x ^ { 2 } + 7 x + 10 } + \frac { - 3 - 2 x } { x ^ { 2 } + 7 x + 10 }x2+7x+103x+5​+x2+7x+10−3−2x​

Find the function value. - f(x)=x−47x+12;f(−2)f ( x ) = \frac { x - 4 } { 7 x + 12 } ; f ( - 2 )f(x)=7x+12x−4​;f(−2)

Simplify the complex fraction. - 6x−1−4y−1−5x−2−4y−2\frac { 6 x ^ { - 1 } - 4 y ^ { - 1 } } { - 5 x ^ { - 2 } - 4 y ^ { - 2 } }−5x−2−4y−26x−1−4y−1​

Add. Simplify the result, if possible. - 78x2+58x2\frac { 7 } { 8 x ^ { 2 } } + \frac { 5 } { 8 x ^ { 2 } }8x27​+8x25​

Multiply as indicated. - 2x24⋅8x3\frac { 2 x ^ { 2 } } { 4 } \cdot \frac { 8 } { x ^ { 3 } }42x2​⋅x38​

Find the domain of the rational function. - f(x)=x−35−xf ( x ) = \frac { x - 3 } { 5 - x }f(x)=5−xx−3​

Simplify the complex fraction. - 9+3xx4+112\frac { 9 + \frac { 3 } { x } } { \frac { x } { 4 } + \frac { 1 } { 12 } }4x​+121​9+x3​​

Divide as indicated. - x2+15x+54x2+16x+63÷x2+6xx2+17x+70\frac { x ^ { 2 } + 15 x + 54 } { x ^ { 2 } + 16 x + 63 } \div \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 17 x + 70 }x2+16x+63x2+15x+54​÷x2+17x+70x2+6x​

Find the least common denominator of the rational expressions. - 1−3x\frac { 1 } { - 3 x }−3x1​ and 2x2+7x\frac { 2 } { x ^ { 2 } + 7 x }x2+7x2​

Add. Simplify the result, if possible. - 413x+613x\frac { 4 } { 13 x } + \frac { 6 } { 13 x }13x4​+13x6​

Simplify the rational expression. If the rational expression cannot be simplified, so state. - 7x2+21x311x+33x2\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 11 x + 33 x ^ { 2 } }11x+33x27x2+21x3​

Find the least common denominator of the rational expressions. - 3x\frac { 3 } { x }x3​ and 3x+6\frac { 3 } { x + 6 }x+63​