Question 1

Which of the following best describes the domain of the rational function $$R(x) = \frac{x^2$ - 4}{x - 2}$$?

Accepted Answer

All real numbers except $$x = 2$$

Question 2

Given the rational function $$H(x) = \frac{1}{x^2$}$$, what happens to $$H(x) as$$ $$x$$ approaches $$0$$ from the right?

Accepted Answer

$$H(x)$$ approaches $$\infty$$

Question 3

For the rational function $$R(x) = \frac{x - 1}{x^2$ - 9}$$, which of the following statements is true about its vertical asymptotes?

Accepted Answer

Vertical asymptotes at $$x = -3$$ and $$x = 3$$

Question 4

A rational function $$R(x) = \frac{2x^2$ - 2x}{x^2$ + x - 6}$$ has a horizontal asymptote. What is its equation?

Accepted Answer

$$y = 2$$

Question 5

If the degree of the numerator of a rational function is one more than the degree of the denominator, what type of asymptote does the function have?

Accepted Answer

Oblique (slant) asymptote

Question 6

Consider $$R(x) = \frac{x^2$ - 4}{x}. $$What is the equation of its oblique asymptote?

Accepted Answer

$$y = x$$

Question 7

For the rational function $$R(x) = \frac{x^4$ + 4}{x^2$}$$, what is the end behavior of the graph as $$x 	o \infty$$?

Accepted Answer

The graph approaches $$y = x^2$

Question 8

A rational function $$R(x)$$ has a vertical asymptote at $$x = c$$ with even multiplicity. How does the graph behave near $$x = c$$?

Accepted Answer

The graph approaches $$\infty or$$ $$-\infty on $$both sides of $$x = c$$

Question 9

Which of the following rational functions has a hole at $$x = 3$$?

Accepted Answer

$$R(x) = \frac{x^2$ - 9}{x^2$ - 6x + 9}$$

Question 10

Given $$R(x) = \frac{2x^3$ + 3x^2$ + 2}{x^2$ + x + 1}$$, what is the equation of its oblique asymptote?

Accepted Answer

$$y = 2x + 1$$

Question 11

For the rational function $$Q(x) = \frac{x^4$ - 16}{5x^2$ - 7x - 6}$$, what is the vertical asymptote?

Accepted Answer

$$x = -\frac{6}{5}$$

Question 12

Which statement about horizontal asymptotes is correct?

Accepted Answer

A graph may intersect its horizontal asymptote

Question 13

If a rational function simplifies to a linear function, what can be said about its asymptotes?

Accepted Answer

It does not have any asymptotes

Question 14

For $$R(x) = \frac{x^2$ - 4}{x}$$, which of the following is true about its x-intercepts?

Accepted Answer

The graph crosses the x-axis at $$x = -2$$ and $$x = 2$$

Question 15

A rational function $$R(x)$$ has a vertical asymptote at $$x = 0$$ with odd multiplicity. What is the behavior of the graph near $$x = 0$$?

Accepted Answer

The graph approaches $$\infty on $$one side and $$-\infty on $$the other

Question 16

For $$R(x) = \frac{x^2$ - 4}{x}$$, what is the domain?

Accepted Answer

All real numbers except $$x = 0$$

Question 17

Which of the following rational functions has no vertical asymptotes?

Accepted Answer

$$R(x) = \frac{x^3$ - 2}{x^3$ + 2}$$

Question 18

For $$R(x) = \frac{x^2$ - 4}{x}$$, does the graph intersect its oblique asymptote $$y = x$$?

Accepted Answer

No, the graph does not intersect $$y = x$$

Question 19

For $$R(x) = \frac{x^4$ + 4}{x^2$}$$, what is the vertical asymptote and its multiplicity?

Accepted Answer

Vertical asymptote at $$x = 0$$ with even multiplicity

Study Notes: Properties and Graphs of Rational Functions

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