$$q(3) = 0$$ and $$p(3) \neq 0$$

Question 1

Which of the following best describes the domain of the function $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$(-\infty, 0) \cup (0, \infty)$$

Question 2

What is the equation of the vertical asymptote for the function $$f(x) = \frac{2}{x+1}$$?

Accepted Answer

$$x = -1$$

Question 3

Given the rational function $$f(x) = \frac{x^2$ + 1}{x - 2}$$, what is the equation of its oblique (slant) asymptote?

Accepted Answer

$$y = x + 2$$

Question 4

Which of the following statements is TRUE about the graph of $$f(x) = \frac{1}{x^2$}$$?

Accepted Answer

It is symmetric with respect to the y-axis.

Question 5

A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has a vertical asymptote at $$x = 3. $$Which of the following must be true?

Accepted Answer

$$q(3) = 0$$ and $$p(3) 
eq 0$$

Question 6

For the function $$f(x) = \frac{2x + 1}{x - 3}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 7

Which transformation describes how the graph of $$f(x) = \frac{1}{x^2$} is $$changed to $$g(x) = \frac{1}{(x+2)^2$} - 1$$?

Accepted Answer

Shifted 2 units left and 1 unit down

Question 8

If the degree of the numerator of a rational function is less than the degree of the denominator, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 9

A parking ramp admits 3.2 vehicles per minute, and vehicles arrive at 2.6 vehicles per minute. What is the traffic intensity $$x$$?

Accepted Answer

$$x = \frac{2.6}{3.2}$$

Question 10

For the function $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what is the nature of the discontinuity at $$x = 2$$?

Accepted Answer

A hole (removable discontinuity)

Question 11

Which of the following rational functions has a horizontal asymptote at $$y = 3$$?

Accepted Answer

$$f(x) = \frac{6x^2$ + 1}{2x^2$ - 5}$$

Question 12

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

Accepted Answer

$$f(x) 	o 0$$

Question 13

If $$f(x) = \frac{x+1}{2x^2$ + 5x - 3}$$, what are the equations of the vertical asymptotes?

Accepted Answer

$$x = -3$$ and $$x = \frac{1}{2}$$

Question 14

Which of the following is true about the graph of $$f(x) = \frac{1}{x}$$?

Accepted Answer

It is an odd function and symmetric with respect to the origin.

Question 15

For the rational function $$f(x) = \frac{3x^2$ - 3x - 6}{x^2$ + 8x + 16}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 3$$

Question 16

Which of the following best describes the behavior of $$f(x) = \frac{1}{x^2$} as$$ $$x$$ approaches 0 from either side?

Accepted Answer

$$f(x) 	o \infty$$

Question 17

Which of the following is a necessary step when graphing a rational function?

Accepted Answer

Find all x- and y-intercepts.

Question 18

If $$f(x) = \frac{2}{x}$$, which transformation would produce the graph of $$y = -\frac{2}{x}$$?

Accepted Answer

Reflection across the x-axis

Question 19

For the rational function $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what is the simplified form (excluding the point of discontinuity)?

Accepted Answer

$$f(x) = x + 2$$

3-05: Study Notes: Rational Functions – Graphs, Asymptotes, and Applications

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