Q: Which of the following best describes the meaning of a horizontal asymptote in a rational function modeling average cost?

It is the minimum possible average cost as production increases.

Question 1

Which of the following rational functions has vertical asymptotes at $x = 1$ and $x = 3$?

Accepted Answer

$f(x) = \dfrac{1}{(x-1)(x-3)}$

Question 2

Given $f(x) = \dfrac{x-2}{x^2-4}$, which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $x = -2$ and a hole at $x = 2$

Question 3

For the function $f(x) = \dfrac{2x}{x^2-4}$, what are the horizontal and vertical asymptotes?

Accepted Answer

Vertical: $x = 2, x = -2$; Horizontal: $y = 0$

Question 4

A government agency allocates $11$ million dollars for pollution removal. The cost to remove $p$ percent of pollutants is $C(p) = \dfrac{20}{100-p}$. What is the maximum percent of pollutants that can be removed with this budget?

Accepted Answer

$p = 80\%$

Question 5

What happens to the cost $C(p) = \dfrac{20}{100-p}$ as $p$ approaches $100$?

Accepted Answer

The cost increases without bound

Question 6

If the average cost per laptop is given by $C(x) = \dfrac{2000}{x} + 1200$, what is the horizontal asymptote and what does it represent?

Accepted Answer

Horizontal asymptote: $y = 1200$; it represents the minimum possible average cost per laptop as production increases.

Question 7

A patient receives a drug with concentration $C(t) = \dfrac{7}{t+2}$ mg/L, where $t$ is hours since administration. How long after administration will the concentration drop to $1$ mg/L?

Accepted Answer

$t = 7$ hours

Question 8

Which of the following best describes the end behavior of $C(x) = \dfrac{2000}{x} + 1200$ as $x$ becomes very large?

Accepted Answer

$C(x)$ approaches $1200$

Question 9

A solution is made by mixing $2$ g/L of glucose with $8$ L of water. The concentration is given by $C(x) = \dfrac{2}{x+2}$, where $x$ is the added volume in liters. What is the concentration after adding $8$ L of water?

Accepted Answer

$0.2$ g/L

Question 10

For the function $f(x) = \dfrac{x^2-4}{x^2-4x+3}$, what are the vertical asymptotes?

Accepted Answer

$x = 1$ and $x = 3$

Question 11

Which of the following rational functions has a hole at $x = 2$ and a vertical asymptote at $x = -2$?

Accepted Answer

$f(x) = \dfrac{x-2}{x^2-4}$

Question 12

If $C(t) = \dfrac{7}{t+2}$, what is the initial concentration of the drug (at $t = 0$)?

Accepted Answer

$3.5$ mg/L

Question 13

Suppose the cost to remove $p$ percent of pollutants is $C(p) = \dfrac{20}{100-p}$. What is the domain of this function in the context of pollution removal?

Accepted Answer

$p < 100$

Question 14

For $f(x) = \dfrac{2x^2-4x}{x^2-4}$, what is the horizontal asymptote?

Accepted Answer

$y = 2$

Question 15

A factory wants the average cost per laptop to be at most $1300$. Using $C(x) = \dfrac{2000}{x} + 1200$, what is the minimum number of laptops that must be produced?

Accepted Answer

$x = 10$

Question 16

Which of the following best describes the meaning of a horizontal asymptote in a rational function modeling average cost?

Accepted Answer

It is the minimum possible average cost as production increases.

Question 17

Given $f(x) = \dfrac{x^2-4}{x^2-4x+3}$, what is the horizontal asymptote?

Accepted Answer

$y = 1$

Question 18

If $C(x) = \dfrac{2000}{x} + 1200$, what is $C(100)$?

Accepted Answer

$1220$

Question 19

A rational function has a vertical asymptote at $x = 3$ and a horizontal asymptote at $y = 2$. Which of the following could be its equation?

Accepted Answer

$f(x) = \dfrac{2x}{x-3}$

E6

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