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

$$p \u003c 100$$

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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