Question 1

Find the function value.
-$$f ( x ) = \frac { x - 4 } { 7 x + 12 } ; f ( - 2 )$$
A) 0
B) 1
C) $- 3$
D) 3

Accepted Answer

D

Question 2

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. 
-Find f(8).
A) 3
B) 8
C) -10.8
D) -8

Accepted Answer

D

Question 3

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

Accepted Answer

The function $f(x) = \frac{2}{x-2}$ is undefined when the denominator is zero, which occurs at $x = 2$. Therefore, the domain excludes $x = 2$, making the correct domain $(-\infty, 2) \cup (2, \infty)$.

Question 4

Find the domain of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 8 x + 12 }$$
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 $f : ( \infty , - 6 ) \cup ( - 6 , - 2 ) \cup ( - 2 , \infty )$

Accepted Answer

The domain of $f(x)$ is determined by the values of $x$ for which the denominator $x^2 - 8x + 12$ is not equal to zero. Factoring the denominator gives $(x-2)(x-6)$, which means the denominator is zero at $x=2$ and $x=6$. Therefore, the domain excludes $x=2$ and $x=6$, making the correct domain $(-\infty, 2) \cup (2, 6) \cup (6, \infty)$.

Question 5

Find the domain of the rational function.
-$$f ( x ) = \frac { x - 3 } { 5 - x }$$
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 $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Accepted Answer

The domain of a function is the set of all possible inputs (x-values) for which the function is defined. For the function $f(x) = \frac{x - 3}{5 - x}$, the only restriction is that the denominator cannot be zero, as division by zero is undefined. Setting the denominator equal to zero and solving for x gives $5 - x = 0$, which simplifies to $x = 5$. Therefore, the function is defined for all real numbers except $x = 5$, which corresponds to the domain $(-\infty, 5) \cup (5, \infty)$.

Question 6

Find the function value.
-$$f ( x ) = \frac { x - 3 } { 2 x } ; f ( - 3 )$$
A) $\frac { 1 } { 2 }$
B) 2
C) 1
D) $- 3$

Accepted Answer

To find $f(-3)$, substitute $x = -3$ into the function: $f(-3) = \frac{-3 - 3}{2(-3)} = \frac{-6}{-6} = 1$.

Question 7

Solve the problem.
-Suppose a cost-benefit model is given by $y = \frac { 5.3 x } { 100 - x }$, where $y$ is the cost in thousands of dollars for removing $x$ percent of a given pollutant. Find the cost for removing $60 \%$ of the pollutant.
A) $\$ 79.5$ thousands
B) $\$ 9.2$ thousands
C) $\$ 8$ thousands
D) $\$ 2$ thousands

Accepted Answer

The answer of Solve the problem.
-Suppose a cost-benefit model is...

Question 8

Find the domain of the rational function.
-$$f ( x ) = \frac { x - 3 } { 6 }$$
A) domain of $\mathrm { f } : ( \infty , \infty )$
B) domain of $f : ( \infty , - 3 ) \cup ( - 3 , \infty )$
C) domain of $f : ( \infty , 3 ) \cup ( 3 , \infty )$
D) domain of $\mathrm { f } : ( \infty , 0 ) \cup ( 0 , \infty )$

Accepted Answer

The function $f(x) = \frac{x - 3}{6}$ is defined for all real numbers because there are no values of $x$ that make the denominator zero or result in an undefined expression.

Question 9

Solve the problem.
-A drug is injected into a patient and the concentration of the drug is monitored. The drug's concentration, $C ( t )$, in milligrams after $t$ hours is modeled by $C ( t ) = \frac { 6 t } { 3 t ^ { 2 } + 2 }$. Estimate the drug's concentration after 4 hours. (Round to the nearest hundredth.)
A) $0.48 \mathrm { mg }$
B) $1.71 \mathrm { mg }$
C) $0.35 \mathrm { mg }$
D) $1.58 \mathrm { mg }$

Accepted Answer

The correct answer is A because $C ( 4 ) = \frac { 6 ( 4 ) } { 3 ( 4 ) ^ { 2 } + 2 } = \frac { 24 } { 46 } \approx 0.48$.

Question 10

Solve the problem.
-To calculate the drug dosage for a child, a pharmacist may use the formula $\mathrm { d } ( \mathrm { x } ) = \frac { \mathrm { Dx } } { \mathrm { x } + 10 } , 0 \leq \mathrm { x } \leq 12$. The child's age is $x$ and the adult dosage is D. What is the dosage for an 3 -year old child if the adult dosage is 40 mg? (Round to the nearest tenth.)
A) $3.1 \mathrm { mg }$
B) $12.0 \mathrm { mg }$
C) $9.2 \mathrm { mg }$
D) $40.0 \mathrm { mg }$

Accepted Answer

The answer of Solve the problem.
-To calculate the drug dosage...

Question 11

Find the domain of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } + 9 x + 14 }$$
A) domain of $f : ( \infty , - 3 ) \cup ( - 3,3 ) \cup ( 3 , \infty )$
B) domain of $f : ( \infty , 2 ) \cup ( 2,7 ) \cup ( 7 , \infty )$
C) domain of $f : ( \infty , - 7 ) \cup ( - 7 , - 2 ) \cup ( - 2 , \infty )$
D) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Accepted Answer

The domain of a function is the set of all possible inputs (x-values) for which the function is defined. For the function $f(x) = \frac{x^2 - 9}{x^2 + 9x + 14}$, the only restrictions on the domain come from the denominator, since division by zero is undefined. The denominator is $x^2 + 9x + 14$, which can be factored as $(x + 7)(x + 2)$. Therefore, the function is undefined when $x = -7$ or $x = -2$, leading to the domain being all real numbers except $x = -7$ and $x = -2$, which matches choice C: $(-\infty, -7) \cup (-7, -2) \cup (-2, \infty)$.

Question 12

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

Accepted Answer

Horizontal asymptotes are of the form y = k, where k is a constant. This means options involving x = k are incorrect, leaving us with options that describe horizontal lines, which are y = k.

Question 13

Find the function value.
-$$f ( x ) = \frac { x ^ { 3 } - 2 } { x ^ { 2 } - 2 } ; f ( 2 )$$
A) $\frac { 3 } { 2 }$
B) 4
C) 1
D) 3

Accepted Answer

The answer of Find the function value.
-\[f ( x )...

Question 14

Find the function value.
-$$f ( x ) = \frac { x ^ { 2 } + 4 } { x ^ { 3 } - 7 x } ; f ( - 5 )$$
A) $- \frac { 5 } { 18 }$
B) $- \frac { 29 } { 90 }$
C) $- \frac { 29 } { 132 }$
D) $- \frac { 29 } { 125 }$

Accepted Answer

To find $f(-5)$, substitute $x = -5$ into the function: $$f(-5) = \frac{(-5)^2 + 4}{(-5)^3 - 7(-5)} = \frac{25 + 4}{-125 + 35} = \frac{29}{-90} = -\frac{29}{90}$$.

Question 15

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. 
-Find f(-2).
A) 1
B) 0
C) -2
D) 2

Accepted Answer

The answer of The graph of a rational function, f,...

Question 16

Find the domain of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } - 81 } { x ^ { 2 } + 7 x - 18 }$$
A) domain of $f : ( \infty , - 9 ) \cup ( - 9,9 ) \cup ( 9 , \infty )$
B) domain of $f : ( \infty , - 9 ) \cup ( - 9,2 ) \cup ( 2 , \infty )$
C) domain of $f : ( \infty , - 2 ) \cup ( - 2,9 ) \cup ( 9 , \infty )$
D) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Accepted Answer

The domain of $f(x)$ is all real numbers except those that make the denominator equal to zero. Factoring the denominator $x^2 + 7x - 18 = (x + 9)(x - 2)$, we find that the denominator is zero when $x = -9$ or $x = 2$. Therefore, the domain excludes -9 and 2, making option B correct.

Question 17

Find the function value.
-$$f ( x ) = \frac { x + 6 } { 14 x - 13 } ; f ( - 2 )$$
A) $\frac { 4 } { 41 }$
B) $\frac { 4 } { 71 }$
C) $- \frac { 4 } { 15 }$
D) $- \frac { 4 } { 41 }$

Accepted Answer

To find $f(-2)$, substitute $x = -2$ into the function: $f(-2) = \frac{-2 + 6}{14(-2) - 13} = \frac{4}{-28 - 13} = \frac{4}{-41} = -\frac{4}{41}$.

Question 18

Find the domain of the rational function.
-$$f ( x ) = \frac { 4 } { x + 4 }$$
A) domain of $f : ( \infty , - 4 ) \cup ( - 4 , \infty )$
B) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$
C) domain of $\mathrm { f } : ( \infty , 4 ) \cup ( 4 , \infty )$
D) domain of $\mathrm { f } : ( \infty , \infty )$

Accepted Answer

The domain of a function includes all the values for which the function is defined. For the function $f(x) = \frac{4}{x + 4}$, it is undefined when the denominator equals zero, which occurs when $x + 4 = 0$ or $x = -4$. Therefore, the domain excludes $x = -4$ and includes all other real numbers, which is represented by $(-\infty, -4) \cup (-4, \infty)$.

Question 19

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question. 
-Find all vertical asymptotes of the graph.
A) x = 4, x = -4
B) x = -6
C) y = 4, y = -4
D) y = -6

Accepted Answer

Vertical asymptotes occur where the function is undefined and typically result from division by zero in the function's formula. They are always of the form x = a, where "a" is a constant, indicating that the asymptotes are vertical lines. Options C and D incorrectly describe horizontal lines (y = constant), which are not vertical asymptotes.

Question 20

Solve the problem.
-A company that produces radios has costs given by the function C(x) = 25x + 25,000, where x is the number of radios manufactured and C(x) is measured in dollars. The average cost to manufacture each radio is given by $\bar { C } ( x ) = \frac { 25 x + 25,000 } { x }$. Find $\bar { C } ( 50 )$. (Round to the nearest dollar, if necessary.)
A) $\$ 75$
B) $\$ 525$
C) $\$ 74$
D) $\$ 535$

Accepted Answer

The answer of Solve the problem.
-A company that produces radios...

Quiz 6: Rational Expressions, Functions, and Equations

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

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

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

Solve the problem. -Suppose a cost-benefit model is given by $y = \frac { 5.3 x } { 100 - x }$ , where $y$ is the cost in thousands of dollars for removing $x$ percent of a given pollutant. Find the cost for removing $60 \%$ of the pollutant.

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.

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

Find the function value. - $f ( x ) = \frac { x ^ { 2 } + 4 } { x ^ { 3 } - 7 x } ; f ( - 5 )$

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

Find the function value. - $f ( x ) = \frac { x + 6 } { 14 x - 13 } ; f ( - 2 )$

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

Quiz 6: Rational Expressions, Functions, and Equations

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)

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

Find the domain of the rational function. - f(x) =2x−2f ( x ) = \frac { 2 } { x - 2 }f(x) =x−22​

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​

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

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)

Solve the problem. -Suppose a cost-benefit model is given by y=5.3x100−xy = \frac { 5.3 x } { 100 - x }y=100−x5.3x​ , where yyy is the cost in thousands of dollars for removing xxx percent of a given pollutant. Find the cost for removing 60%60 \%60% of the pollutant.

Find the domain of the rational function. - f(x) =x−36f ( x ) = \frac { x - 3 } { 6 }f(x) =6x−3​

Find the domain of the rational function. - f(x) =x2−9x2+9x+14f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } + 9 x + 14 }f(x) =x2+9x+14x2−9​

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.

Find the function value. - f(x) =x3−2x2−2;f(2) f ( x ) = \frac { x ^ { 3 } - 2 } { x ^ { 2 } - 2 } ; f ( 2 ) f(x) =x2−2x3−2​;f(2)

Find the function value. - f(x) =x2+4x3−7x;f(−5) f ( x ) = \frac { x ^ { 2 } + 4 } { x ^ { 3 } - 7 x } ; f ( - 5 ) f(x) =x3−7xx2+4​;f(−5)