Question 1

Simplify the rational expression. $\frac { 6 - x } { 10 x - 60 }$
A) $- \frac { 1 } { 4 } , x \neq 6$
B) $- \frac { 1 } { 10 } , x \neq 6$
C) $6 , x \neq 6$
D) $10 , x \neq 6$
E) $\frac { 1 } { 6 } , x \neq 6$

Accepted Answer

We can simplify the expression by factoring out a common factor of 2 from both the numerator and the denominator, then simplifying further:
$$\frac{6-x}{10x-60} = \frac{-(x-6)}{10(x-6)} = \frac{-(x-6)}{10}\cdot \frac{1}{x-6} = -\frac{1}{10}$$
However, we must also note that $x 
eq 6$ because that would result in division by zero. Thus, the best choice is B, which simplifies the expression to $-\frac{1}{10}$ and includes the necessary restriction on $x$.

Question 2

The average cost $\bar C$ for a manufacturer to produce x units of a product is given by $\bar { C } = \frac { 1.25 x + 3510 } { x }$ . Describe the domain.
A)all $x > 0$
B)all $x > 1.25$
C)all $x \neq 3510$
D)all integers $> 0$
E)all $x \neq 0$

Accepted Answer

D

Question 3

Find the domain of the rational function. $f ( y ) = \frac { y - 10 } { y ( y + 2 ) }$
A) $( - \infty , 10 ) \cup ( 10 , \infty )$
B) $( - \infty , - 2 ) \cup ( - 2,10 ) \cup ( 10 , \infty )$
C) $( - \infty , - 2 ) \cup ( - 2 , \infty )$
D) $( - \infty , \infty )$
E) $( - \infty , - 2 ) \cup ( - 2,0 ) \cup ( 0 , \infty )$

Accepted Answer

The domain of a rational function excludes values that make the denominator zero. Here, the denominator is $y(y+2)$, which is zero when $y = 0$ or $y = -2$. Thus, the domain is all real numbers except $y = 0$ and $y = -2$.

Question 4

Simplify the rational expression. $\frac { x ^ { 2 } + 8 x y - 9 y ^ { 2 } } { x ^ { 2 } + 10 x y + 9 y ^ { 2 } }$
A) $\frac { x - y } { x + y } , x \neq - 9 y$
B) $\frac { x - y } { x + 9 y } , x \neq y$
C) $\frac { x + y } { x - 9 y } , x \neq 9 y$
D) $\frac { x + 9 y } { x + y } , x \neq - y$
E) $\frac { x - 9 y } { x + 9 y } , x \neq - 9 y$

Accepted Answer

Factor the numerator and denominator:
\begin{align*}
\frac{x^2+8xy-9y^2}{x^2+10xy+9y^2}&=\frac{(x- y)(x+9y)}{(x+y)(x+9y)}\
&=\frac{x-y}{x+y},\quad	ext{for } x
eq -9y
\end{align*}

Question 5

Simplify the rational expression. $\frac { y ^ { 3 } - 9 y } { y ^ { 2 } + 5 y - 24 }$
A) $\frac { y - 3 } { y - 8 } , y \neq 8$
B) $\frac { y + 3 } { 8 } , y \neq - 8 , y \neq 3$
C) $\frac { y ( y + 3 ) } { y + 8 } , y \neq 3$
D) $- \frac { y - 9 } { 18 } , y \neq 0 , y \neq 8$
E) $\frac { y - 3 } { y + 3 } , y \neq - 3$

Accepted Answer

The answer of Simplify the rational expression. \(\frac { y...

Question 6

Find the domain of the rational function. $f ( x ) = \frac { x ^ { 2 } + 4 } { 6 }$
A) $( - \infty , \infty )$
B) $( - \infty , 4 ) \cup ( 4,6 ) \cup ( 6 , \infty )$
C) $( - \infty , - 4 ) \cup ( - 4,4 ) \cup ( 4 , \infty )$
D) $( - \infty , 6 ) \cup ( 6 , \infty )$
E) $( - \infty , 4 ) \cup ( 4 , \infty )$

Accepted Answer

The given rational function is defined for all real numbers because the denominator is a constant and never equals zero. Therefore, the domain of the function is all real numbers, which is represented by choice A.

Question 7

Simplify the rational expression. $\frac { 7 x y ^ { 2 } } { x y ^ { 2 } + x }$
A) $\frac { 7 } { x + 1 } , x \neq - 1$
B) $\frac { 7 y ^ { 2 } } { x y ^ { 2 } + 1 } , y \neq 0$
C) $\frac { 7 y ^ { 2 } } { y ^ { 2 } + 1 } , x \neq 0$
D) $\frac { 7 y ^ { 2 } } { y ^ { 2 } + x } , y \neq 0$
E) $\frac { 7 } { x } , x \neq 0$

Accepted Answer

The answer of Simplify the rational expression. \(\frac { 7...

Question 8

Simplify the rational expression. $\frac { 3 x y + 5 x ^ { 2 } y ^ { 2 } } { x y ^ { 4 } }$
A) $\frac { 3 x + 5 y } { y ^ { 3 } } , x \neq 0 , y \neq 0$
B) $\frac { 3 y + 5 x ^ { 2 } } { y ^ { 2 } } , x \neq 0 , y \neq 0$
C) $\frac { 3 + 5 x ^ { 2 } y ^ { 2 } } { y ^ { 3 } } , y \neq 0$
D) $\frac { x ( 3 y + 5 ) } { y ^ { 2 } } , y \neq 0$
E) $\frac { 3 + 5 x y } { y ^ { 3 } } , x \neq 0 , y \neq 0$

Accepted Answer

The answer of Simplify the rational expression. \(\frac { 3...

Question 9

Find the domain of the rational function. $h ( t ) = \frac { 10 t ^ { 2 } } { t ^ { 2 } - 4 t - 45 }$
A) $( - \infty , 0 ) \cup ( 0 , \infty )$
B) $( - \infty , \infty )$
C) $( - \infty , - 5 ) \cup ( - 5,0 ) \cup ( 0 , \infty )$
D) $( - \infty , 0 ) \cup ( 0,9 ) \cup ( 9 , \infty )$
E) $( - \infty , - 5 ) \cup ( - 5,9 ) \cup ( 9 , \infty )$

Accepted Answer

The domain of a rational function is all real numbers except where the denominator equals zero. To find these points for $h(t) = \frac{10t^2}{t^2 - 4t - 45}$, set the denominator equal to zero and solve for $t$: $t^2 - 4t - 45 = 0$. Factoring the quadratic equation gives $(t - 9)(t + 5) = 0$, so $t = 9$ and $t = -5$ are the points where the function is undefined. Thus, the domain is all real numbers except $t = 9$ and $t = -5$, which corresponds to $( - \infty , - 5 ) \cup ( - 5,9 ) \cup ( 9 , \infty )$.

Question 10

Simplify the rational expression. $\frac { 30 x ^ { 2 } + 11 x - 7 } { 100 x ^ { 2 } - 49 }$
A) $\frac { 3 x - 1 } { 10 x + 7 } , x \neq \frac { 10 } { 7 }$
B) $\frac { 10 x - 7 } { 3 } , x \neq 0$
C) $\frac { 7 - 10 x } { 3 } , x \neq 0$
D) $\frac { 10 x + 7 } { 10 x - 7 } , x \neq \frac { 7 } { 10 }$
E) $\frac { 3 x - 1 } { 10 x - 7 } , x \neq \frac { 7 } { 10 }$

Accepted Answer

We can factor both the numerator and denominator to simplify the expression.
\begin{align*}
\frac{30x^2+11x-7}{100x^2-49} &= \frac{(6x-1)(5x+7)}{(10x+7)(10x-7)} \
&= \frac{6x-1}{10x-7} \cdot \frac{5x+7}{10x+7} \
&= \frac{3x-1}{5x+7} \cdot \frac{5x+7}{10x+7} \
&= \frac{3x-1}{10x-7}, 	ext{ for } x 
eq \frac{7}{10}
\end{align*}
Therefore, the best choice is E.

Question 11

Find the domain of the rational function. $f ( x ) = \frac { 9 } { x - 5 }$
A) $( - \infty , 5 ) \cup ( 5 , \infty )$
B) $( - \infty , 9 ) \cup ( 9 , \infty )$
C) $( - \infty , - 5 ) \cup ( - 5,5 ) \cup ( 5 , \infty )$
D) $( - \infty , \infty )$
E) $( - \infty , 5 ) \cup ( 5,9 ) \cup ( 9 , \infty )$

Accepted Answer

The denominator cannot be equal to zero, so the function is undefined at $x = 5$. Therefore, the domain is all real numbers except $5$, which is represented by the interval notation $( - \infty , 5 ) \cup ( 5 , \infty )$.

Question 12

Find the domain of the rational function. $f ( x ) = \frac { x } { x ^ { 2 } - 81 }$
A) $( - \infty , 0 ) \cup ( 0,9 ) \cup ( 9 , \infty )$
B) $( - \infty , - 9 ) \cup ( - 9,9 ) \cup ( 9 , \infty )$
C) $( - \infty , 9 ) \cup ( 9 , \infty )$
D) $( - \infty , 0 ) \cup ( 0 , \infty )$
E) $( - \infty , \infty )$

Accepted Answer

The denominator $x^2 - 81$ factors to $(x + 9)(x - 9)$, so the function is undefined at $x = -9$ and $x = 9$. The domain includes all other real numbers, hence $( - \infty , - 9 ) \cup ( - 9,9 ) \cup ( 9 , \infty )$.

Question 13

Find the domain of the rational function. $h ( x ) = \frac { 6 x } { x ^ { 2 } + 4 }$
A) $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$
B) $( - \infty , \infty )$
C) $( - \infty , 6 ) \cup ( 6 , \infty )$
D) $( - \infty , 2 ) \cup ( 2,6 ) \cup ( 6 , \infty )$
E) $( - \infty , 0 ) \cup ( 0 , \infty )$

Accepted Answer

The denominator $x^2 + 4$ is never zero for any real value of $x$, because $x^2$ is always non-negative, making $x^2 + 4 > 0$ for all real $x$. Therefore, the domain is all real numbers.

Question 14

Simplify the rational expression. $\frac { 3 x ^ { 2 } - 10 x - 25 } { 40 - 3 x - x ^ { 2 } }$
A) $\frac { x + 5 } { x - 5 } , x \neq 8$
B) $- \frac { 3 x + 5 } { x + 8 } , x \neq 5$
C) $\frac { x - 35 } { 37 - x } , x \neq 0$
D) $- \frac { 3 x + 5 } { x - 5 } , x \neq - 5$
E) $\frac { x - 5 } { x + 8 } , x \neq - 8$

Accepted Answer

The answer of Simplify the rational expression. \(\frac { 3...

Question 15

Find the ratio of the area of the shaded portion to the total area of the figure. A = 5 and B = 6.

A) $\frac { x } { x + 6 }$
B) 5
6
C) $\frac { x } { x + 5 }$
D) $\frac { x } { x + 1 }$
E) $\frac { x ( x + 5 ) } { x + 6 }$

Accepted Answer

The answer of Find the ratio of the area of...

Question 16

A machine shop has a setup cost of $3000 for the production of a new product. The cost, C, of labor and material for producing each unit is $10.75. The average cost function for producing x units is given by $\bar { C } = C / x$ . Determine the domain of the average cost function.
A)all real numbers
B)all $x > 10.75$
C)all $x \neq 0$
D)all integers $x > 0$
E)all integers $x > 3000$

Accepted Answer

The setup cost of $3000 is a one-time fixed cost and is not affected by the number of units produced. The cost of labor and material for each unit is $10.75. Therefore, the total cost of producing x units is given by 
C(x) = 3000 + 10.75x 
The average cost function is given by 
$\bar { C } = \frac { C(x) }{ x } = \frac { 3000 }{ x } + 10.75$
The domain of the average cost function is all positive integers greater than 0, because the average cost function is not defined for x=0. Therefore, the best choice is D, all integers x >0.

Question 17

Evaluate the rational function $f ( x ) = \frac { 8 x } { x + 13 }$ at x=-13 and simplify. If this is not possible, state the reason.
A)0
B)-13
C)4
D)-104
E)undefined $\text { (division by } 0 \text { ) }$

Accepted Answer

At x=-13, we get a division by 0 in the denominator of the rational function. Therefore, the function is undefined at x=-13. The correct answer choice is E.

Question 18

Simplify the rational expression. $\frac { a + 3 } { a ^ { 2 } + 6 a + 9 }$
A) $\frac { 1 } { 7 a + 3 } , a \neq - \frac { 3 } { 7 }$
B) $\frac { 1 } { a ^ { 2 } + 9 } , a \neq 3$
C) $\frac { 1 } { a + 9 } , a \neq - 9$
D) $\frac { 1 } { a + 3 } , a \neq - 3$
E) $\frac { 1 } { 3 a + 9 } , a \neq - \frac { 3 } { 1 }$

Accepted Answer

We can factor the denominator as $(a+3)^2$. Therefore, we have $\frac{a+3}{(a+3)^2}$. Canceling the common factor, we get $\frac{1}{a+3}$, which matches with option D.

Question 19

Simplify the rational expression. $\frac { 9 x ^ { 2 } - 36 x } { 72 x ^ { 2 } }$
A) $\frac { 9 x ^ { 2 } - 4 } { 8 x }$
B) $\frac { x - 4 } { 8 x }$
C) $\frac { x - 9 } { 8 }$
D) $\frac { 1 - 4 x } { 8 x }$
E) $\frac { 1 - 36 x } { 8 }$

Accepted Answer

The greatest common factor of the numerator is $9x$, so factor out $9x$ from the numerator and denominator: \begin{align*}
\frac{9x( x-4)}{72x^2}&=\frac{x-4}{8x} &&	ext{(simplify)}\
&=\boxed{	extbf{(B) }\frac{x-4}{8x}}.
\end{align*}

Question 20

Evaluate the rational function $f ( x ) = \frac { 7 x } { x + 5 }$ at x=-6 and simplify. If this is not possible, state the reason.
A)85
B)55
C)6
D)42
E)undefined $( \text { division by } 0 \text { ) }$

Accepted Answer

The answer of Evaluate the rational function \(f ( x...

Quiz 6: Rational Expressions, Equations, and Functions

Simplify the rational expression. 6−x10x−60\frac { 6 - x } { 10 x - 60 }10x−606−x​

The average cost Cˉ\bar CCˉ for a manufacturer to produce x units of a product is given by Cˉ=1.25x+3510x\bar { C } = \frac { 1.25 x + 3510 } { x }Cˉ=x1.25x+3510​ . Describe the domain.

Find the domain of the rational function. f(y) =y−10y(y+2) f ( y ) = \frac { y - 10 } { y ( y + 2 ) }f(y) =y(y+2) y−10​

Simplify the rational expression. x2+8xy−9y2x2+10xy+9y2\frac { x ^ { 2 } + 8 x y - 9 y ^ { 2 } } { x ^ { 2 } + 10 x y + 9 y ^ { 2 } }x2+10xy+9y2x2+8xy−9y2​

Simplify the rational expression. y3−9yy2+5y−24\frac { y ^ { 3 } - 9 y } { y ^ { 2 } + 5 y - 24 }y2+5y−24y3−9y​

Find the domain of the rational function. f(x) =x2+46f ( x ) = \frac { x ^ { 2 } + 4 } { 6 }f(x) =6x2+4​

Simplify the rational expression. 7xy2xy2+x\frac { 7 x y ^ { 2 } } { x y ^ { 2 } + x }xy2+x7xy2​

Simplify the rational expression. 3xy+5x2y2xy4\frac { 3 x y + 5 x ^ { 2 } y ^ { 2 } } { x y ^ { 4 } }xy43xy+5x2y2​

Find the domain of the rational function. h(t) =10t2t2−4t−45h ( t ) = \frac { 10 t ^ { 2 } } { t ^ { 2 } - 4 t - 45 }h(t) =t2−4t−4510t2​

Simplify the rational expression. 30x2+11x−7100x2−49\frac { 30 x ^ { 2 } + 11 x - 7 } { 100 x ^ { 2 } - 49 }100x2−4930x2+11x−7​

Find the domain of the rational function. f(x) =9x−5f ( x ) = \frac { 9 } { x - 5 }f(x) =x−59​

Find the domain of the rational function. f(x) =xx2−81f ( x ) = \frac { x } { x ^ { 2 } - 81 }f(x) =x2−81x​

Find the domain of the rational function. h(x) =6xx2+4h ( x ) = \frac { 6 x } { x ^ { 2 } + 4 }h(x) =x2+46x​

Simplify the rational expression. 3x2−10x−2540−3x−x2\frac { 3 x ^ { 2 } - 10 x - 25 } { 40 - 3 x - x ^ { 2 } }40−3x−x23x2−10x−25​

Find the ratio of the area of the shaded portion to the total area of the figure. A = 5 and B = 6.

A machine shop has a setup cost of $3000 for the production of a new product. The cost, C, of labor and material for producing each unit is $10.75. The average cost function for producing x units is given by Cˉ=C/x\bar { C } = C / xCˉ=C/x . Determine the domain of the average cost function.

Evaluate the rational function f(x) =8xx+13f ( x ) = \frac { 8 x } { x + 13 }f(x) =x+138x​ at x=-13 and simplify. If this is not possible, state the reason.

Simplify the rational expression. a+3a2+6a+9\frac { a + 3 } { a ^ { 2 } + 6 a + 9 }a2+6a+9a+3​

Simplify the rational expression. 9x2−36x72x2\frac { 9 x ^ { 2 } - 36 x } { 72 x ^ { 2 } }72x29x2−36x​

Evaluate the rational function f(x) =7xx+5f ( x ) = \frac { 7 x } { x + 5 }f(x) =x+57x​ at x=-6 and simplify. If this is not possible, state the reason.

Simplify the rational expression. $\frac { 6 - x } { 10 x - 60 }$

The average cost $\bar C$ for a manufacturer to produce x units of a product is given by $\bar { C } = \frac { 1.25 x + 3510 } { x }$ . Describe the domain.

Find the domain of the rational function. $f ( y ) = \frac { y - 10 } { y ( y + 2 ) }$

Simplify the rational expression. $\frac { x ^ { 2 } + 8 x y - 9 y ^ { 2 } } { x ^ { 2 } + 10 x y + 9 y ^ { 2 } }$

Simplify the rational expression. $\frac { y ^ { 3 } - 9 y } { y ^ { 2 } + 5 y - 24 }$

Find the domain of the rational function. $f ( x ) = \frac { x ^ { 2 } + 4 } { 6 }$

Simplify the rational expression. $\frac { 7 x y ^ { 2 } } { x y ^ { 2 } + x }$

Simplify the rational expression. $\frac { 3 x y + 5 x ^ { 2 } y ^ { 2 } } { x y ^ { 4 } }$

Find the domain of the rational function. $h ( t ) = \frac { 10 t ^ { 2 } } { t ^ { 2 } - 4 t - 45 }$

Simplify the rational expression. $\frac { 30 x ^ { 2 } + 11 x - 7 } { 100 x ^ { 2 } - 49 }$

Find the domain of the rational function. $f ( x ) = \frac { 9 } { x - 5 }$

Find the domain of the rational function. $f ( x ) = \frac { x } { x ^ { 2 } - 81 }$

Find the domain of the rational function. $h ( x ) = \frac { 6 x } { x ^ { 2 } + 4 }$

Simplify the rational expression. $\frac { 3 x ^ { 2 } - 10 x - 25 } { 40 - 3 x - x ^ { 2 } }$

A machine shop has a setup cost of $3000 for the production of a new product. The cost, C, of labor and material for producing each unit is $10.75. The average cost function for producing x units is given by $\bar { C } = C / x$ . Determine the domain of the average cost function.

Evaluate the rational function $f ( x ) = \frac { 8 x } { x + 13 }$ at x=-13 and simplify. If this is not possible, state the reason.

Simplify the rational expression. $\frac { a + 3 } { a ^ { 2 } + 6 a + 9 }$

Simplify the rational expression. $\frac { 9 x ^ { 2 } - 36 x } { 72 x ^ { 2 } }$

Evaluate the rational function $f ( x ) = \frac { 7 x } { x + 5 }$ at x=-6 and simplify. If this is not possible, state the reason.