Question 1

Which of the following is the correct domain of the function $$f(x) = \frac{x^2$ + 8}{x - 2}$$?

Accepted Answer

$$(-\infty, 2) \cup (2, \infty)$$

Question 2

Given the rational function $$f(x) = \frac{x^2$ + 6x}{x^2$ - 8x}$$, which value(s) of $$x$$ are excluded from the domain?

Accepted Answer

$$x = 0$$ and $$x = 8$$

Question 3

As $$x$$ approaches $$2$$ from the right in the function $$f(x) = \frac{x^2$ + 8}{x - 2}$$, what happens to $$f(x)$$?

Accepted Answer

$$f(x) 	o +\infty$$

Question 4

Which of the following best describes the behavior of $$f(x) = \frac{1}{x} as$$ $$x 	o -\infty$$?

Accepted Answer

$$f(x) 	o 0$$

Question 5

For the function $$f(x) = \frac{x^2$ - 4}{x^2$ - 4}$$, what is true about its graph?

Accepted Answer

There is a hole at $$x = 2$$ and $$x = -2$$; elsewhere, $$f(x) = 1$$

Question 6

Find the vertical asymptotes and holes for $$f(x) = \frac{x^2$ - 4}{x^2$ - 4x}.$$

Accepted Answer

Vertical asymptote at $$x = 0$$; hole at $$x = 4$$

Question 7

Which of the following statements about horizontal asymptotes is correct for rational functions?

Accepted Answer

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $$y = 0$$

Question 8

What is the horizontal asymptote of $$f(x) = \frac{3x^2$ + 15}{x^2$ + 1}$$?

Accepted Answer

$$y = 3$$

Question 9

A rational function has a numerator of degree 3 and a denominator of degree 2. What type of asymptote does it have?

Accepted Answer

A slant (oblique) asymptote

Question 10

Find the equation of the slant asymptote for $$f(x) = \frac{x^2$ + 1}{x - 1}.$$

Accepted Answer

$$y = x + 1$$

Question 11

For $$f(x) = \frac{x^3$ - 7x + 5}{x - 2}$$, what is the equation of the slant asymptote?

Accepted Answer

$$y = x^2$ + 2x - 3$$

Question 12

A chemical reaction's rate is modeled by $$R(t) = \frac{2t}{t^2$ + 1}. As $$time $$t$$ increases without bound, what does the reaction rate approach?

Accepted Answer

$$0$$

Question 13

A medical dosage model is given by $$D(x) = \frac{4x}{x^2$ + 4}. $$What is the horizontal asymptote of this function?

Accepted Answer

$$y = 0$$

Question 14

Which of the following steps is NOT part of the standard process for graphing a rational function?

Accepted Answer

Find the minimum value of the function

Question 15

For $$f(x) = \frac{x^2$ - 4}{x^2$ - 9}$$, which of the following is true?

Accepted Answer

Vertical asymptotes at $$x = 3$$ and $$x = -3$$; $$x-$$intercepts at $$x = 2$$ and $$x = -2$$

Question 16

A rational function $$f(x)$$ has a hole at $$x = 5. $$Which of the following must be true?

Accepted Answer

Both the numerator and denominator have a factor of $$(x - 5)$$

Question 17

Suppose a population model is given by $$P(t) = \frac{100t}{t^2$ + 25}. $$What is the long-term behavior of the population as $$t 	o \infty$$?

Accepted Answer

$$P(t) 	o 0$$

Question 18

If $$f(x) = \frac{x^2$ - 25}{x + 5}$$, what is the $$x-$$intercept(s) of the function?

Accepted Answer

$$x = 5$$ and $$x = -5$$

Question 19

Which of the following best describes the end behavior of $$f(x) = \frac{2x^2$ + 1}{x^2$ + 4} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 2$$

Question 20

A rational function $$f(x)$$ has a vertical asymptote at $$x = -3$$ and a hole at $$x = 2. $$Which of the following could be its formula?

Accepted Answer

$$f(x) = \frac{(x - 2)(x + 1)}{(x + 3)(x - 2)}$$

Rational Functions and Their Graphs: Study Notes

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