Question 1

Which of the following is the correct definition of a rational function?

Accepted Answer

A function that can be written as $$f(x) = \frac{P(x)}{Q(x)}$$ where $$P(x)$$ and $$Q(x)$$ are polynomials and $$Q(x) 
eq 0$$

Question 2

Given $$f(x) = \frac{4x}{2-x}$$, what is the vertical asymptote of the function?

Accepted Answer

$$x = 2$$

Question 3

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

Accepted Answer

$$y = 0$$

Question 4

What is the domain of $$f(x) = \frac{x+1}{x^2$-4}$$?

Accepted Answer

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

Question 5

If $$f(x) = \frac{3x^2$-x+2}{x+8}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 3$$

Question 6

Which step should be performed first when graphing a rational function according to the procedure?

Accepted Answer

Determine the asymptotes and draw them as dashed lines

Question 7

For $$f(x) = \frac{x^3$-x^2-4x+5$$}{$$x^2-4$$}$, what type of asymptote exists and how do you find it?

Accepted Answer

Oblique asymptote; divide the numerator by the denominator using long division

Question 8

What happens to $$f(x) = \frac{1}{x} as$$ $$x$$ becomes very large (positive or negative)?

Accepted Answer

$$f(x)$$ approaches $$0$$

Question 9

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

Accepted Answer

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

Question 10

When solving the rational inequality $$\frac{1}{x-1} \geq 1$$, what is the first step?

Accepted Answer

Rewrite the inequality with zero on one side

Question 11

For $$f(x) = \frac{2}{x^2$+1}$$, what is the domain?

Accepted Answer

All real numbers

Question 12

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 13

For $$f(x) = \frac{x-5}{x^2$-4x-5}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 5$$ and $$x = -1$$

Question 14

When graphing a rational function, why is it important to check for symmetry?

Accepted Answer

It helps predict the shape and behavior of the graph

Question 15

For $$f(x) = \frac{x+1}{x^2$-4}$$, does the function cross its horizontal asymptote?

Accepted Answer

No, it does not cross the horizontal asymptote

Question 16

What is the correct procedure for solving a rational inequality using the test-point method?

Accepted Answer

Rewrite the inequality, factor, find zeros, divide the x-axis into intervals, test points, write solution in interval form

Question 17

For $$f(x) = \frac{3x^2$-x}{x^2$-4}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 18

If a rational function has a hole in its graph, what does this indicate?

Accepted Answer

A common factor in the numerator and denominator that cancels out

Question 19

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

Accepted Answer

$$y = 2$$

Question 20

When dividing the numerator and denominator by the highest power of $$x in$$ $$f(x) = \frac{2x-3}{x^2$+1}$$, what is the resulting behavior as $$x$$ approaches infinity?

Accepted Answer

$$f(x)$$ approaches $$0$$

Rational Functions and Inequalities: Asymptotes, Graphing, and Inequality Solutions

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