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 types of asymptotes?

Accepted Answer

Vertical at $x = 2$ and $x = -2$, horizontal at $y = 0$

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

$p = 45\%$ (since $C(p) = 11$ when $p = 45$)

Question 5

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

Accepted Answer

The cost becomes infinitely large

Question 6

If the average cost per laptop is given by $C(x) = \dfrac{2000}{x} + 1200$, where $x$ is the number of laptops produced, what is the horizontal asymptote of $C(x)$?

Accepted Answer

$y = 1200$

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 be $1$ mg/L?

Accepted Answer

$t = 5$ hours$\ (1 = \dfrac{7}{t+2} \implies t+2=7 \implies t=5)$

Question 8

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

Accepted Answer

$x = 2$ and $x = -2$

Question 9

A solution is made by mixing $2$ L of glucose with $x$ L of water. The concentration is $C(x) = \dfrac{2}{x+2}$. What happens to the concentration as $x$ increases?

Accepted Answer

It approaches $0$

Question 10

Given $C(x) = \dfrac{2000}{x} + 1200$, how many laptops must be produced for the average cost to be $1300$?

Accepted Answer

$x = 20$

Question 11

Which of the following best describes the real-world meaning of the horizontal asymptote in $C(x) = \dfrac{2000}{x} + 1200$?

Accepted Answer

The average cost per laptop approaches $1200$ as more laptops are produced

Question 12

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

Accepted Answer

$y = 2$

Question 13

A patient’s blood drug concentration is $C(t) = \dfrac{7}{t+2}$. What is the concentration $5$ hours after administration?

Accepted Answer

$1$ mg/L

Question 14

Which of the following rational functions has a hole at $x = 2$?

Accepted Answer

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

Question 15

If $C(p) = \dfrac{20}{100-p}$, what is the domain of $C(p)$ in the context of removing pollutants?

Accepted Answer

$p < 100$

Question 16

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

Accepted Answer

$y = 3$

E6

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