Question 1

Which of the following best describes the domain of the rational function $$R(x) = \frac{3x^2$ - 5}{x + 2}$$?

Accepted Answer

All real numbers except $$x = -2$$

Question 2

Given $$R(x) = \frac{1}{x^2$ - 25}$$, which values must be excluded from the domain?

Accepted Answer

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

Question 3

A rational function $$R(x) = \frac{p(x)}{q(x)} is $$defined as:

Accepted Answer

A function where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x) 
eq 0$$

Question 4

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

Accepted Answer

$$R(x) = \frac{x+5}{x-3}$$

Question 5

For the function $$H(x) = \frac{1}{x^2$}$$, which of the following statements is true?

Accepted Answer

The graph is symmetric with respect to the y-axis

Question 6

What is the horizontal asymptote of $$R(x) = \frac{3x^4$ - 2x^2$ - x}{x^6$ + 4x^4$ + 8}$$?

Accepted Answer

$$y = 0$$

Question 7

Which of the following rational functions has an oblique (slant) asymptote?

Accepted Answer

$$H(x) = \frac{2x^3$ + 3x^2$ + 2}{x^2$ + x + 1}$$

Question 8

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

Accepted Answer

$$y = 0$$

Question 9

A rational function $$R(x) = \frac{ax^n + \ldots}{bx^n + \ldots}$$, where $$n is $$the same for numerator and denominator, has which horizontal asymptote?

Accepted Answer

$$y = \frac{a}{b}$$

Question 10

Which of the following best describes the end behavior of $$G(x) = \frac{10x^5$ - x^3$ + 2}{x^3$ - 1} as$$ $$x 	o \infty$$?

Accepted Answer

The graph behaves like $$y = 10x^2$

Question 11

Suppose a rational function $$R(x)$$ has a vertical asymptote at $$x = 2$$ with even multiplicity. What happens to the graph near $$x = 2$$?

Accepted Answer

The graph approaches the same infinity (either $$+\infty or$$ $$-\infty) on $$both sides of $$x = 2$$

Question 12

A chemical reaction rate is modeled by $$R(x) = \frac{5x}{x^2$ + 1}. $$What is the horizontal asymptote and what does it represent in this context?

Accepted Answer

$$y = 0$$, representing the rate approaches zero as concentration increases

Question 13

Which of the following rational functions has no vertical asymptotes?

Accepted Answer

$$H(x) = \frac{x+9}{x^2$ + 9}$$

Question 14

A rational function $$R(x) = \frac{x^2$ - 4}{x - 2}$$ simplifies to $$x + 2$$ for $$x 
eq 2. $$What is the domain of $$R(x)$$?

Accepted Answer

All real numbers except $$x = 2$$

Question 15

If a rational function has a numerator of degree 4 and a denominator of degree 2, what can you say about its asymptotes?

Accepted Answer

There are no horizontal or oblique asymptotes

Question 16

A rational function models the spread of a disease: $$D(x) = \frac{100x}{x^2$ + 10}. $$What is the long-term behavior as $$x ($$population) increases?

Accepted Answer

The spread rate approaches zero

Question 17

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

Accepted Answer

$$y = 2x + 1$$

Question 18

If a rational function $$R(x)$$ has a vertical asymptote at $$x = 5$$ with odd multiplicity, what happens to the graph near $$x = 5$$?

Accepted Answer

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

Question 19

A rational function $$R(x) = \frac{9x^2$ - x + 1}{3x^2$ - 2}$$ has which horizontal asymptote?

Accepted Answer

$$y = 3$$

Properties of Rational Functions (Precalculus, Section 4.5)

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