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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