Question 1

Given the rational function $$f(x) = \frac{x^3$ - 4x^2$ + 3x}{3x^2$ - 15x + 12}$$, which of the following statements about its domain is correct?

Accepted Answer

The domain is all real numbers except $$x = 1$$ and $$x = 4$$

Question 2

For the function $$f(x) = \frac{1}{-3x^2$ + 9x}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 3

A chemical reaction rate is modeled by $$f(x) = \frac{2x^3$ - 4x^2$}{x^3$ - 2x^2$ - 3x}. $$What is the value of $$x$$ where the function has a hole?

Accepted Answer

$$x = 0$$

Question 4

For the function $$f(x) = -\frac{3x}{x^3$ + x^2$ - 6x}$$, which of the following is NOT a vertical asymptote?

Accepted Answer

$$x = 0$$

Question 5

A population model is given by $$f(x) = \frac{x^3$ + x^2$ - 6x}{4x^3$ - 4x^2$ - 8x}. $$What is the horizontal asymptote?

Accepted Answer

$$y = \frac{1}{4}$$

Question 6

For $$f(x) = \frac{x-1}{4x-8}$$, what is the $$y-$$intercept?

Accepted Answer

$$y = \frac{1}{8}$$

Question 7

Which of the following rational functions has a hole at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{2x^3$ - 4x^2$}{x^3$ - 2x^2$ - 3x}$$

Question 8

For $$f(x) = \frac{x^3$ - 9x}{4x^2$ + 8x}$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 9

A rational function $$f(x) = -\frac{3}{x^2$ - 2x - 3} is $$used to model a medical dosage. What is the $$y-$$intercept?

Accepted Answer

$$y = 1$$

Question 10

For $$f(x) = \frac{x^2$ - 1}{3x^3$ + 9x^2$ + 6x}$$, which value of $$x$$ creates a hole in the graph?

Accepted Answer

$$x = -1$$

Question 11

Which rational function has a horizontal asymptote at $$y = 2$$?

Accepted Answer

$$f(x) = \frac{2x^3$ - 4x^2$}{x^3$ - 2x^2$ - 3x}$$

Question 12

For $$f(x) = \frac{x^3$ - 4x^2$ + 3x}{4x^2$ + 4x}$$, what is the domain?

Accepted Answer

All real numbers except $$x = -1$$ and $$x = 0$$

Question 13

A rational function $$f(x) = \frac{x^3$ - 9x}{4x^2$ + 8x} is $$used to model a population. What is the domain?

Accepted Answer

All real numbers except $$x = -2$$ and $$x = 0$$

Question 14

For $$f(x) = \frac{x^2$ - 1}{3x^3$ + 9x^2$ + 6x}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 15

Which function has $$x-$$intercepts at $$x = 3$$ and $$x = 1$$?

Accepted Answer

$$f(x) = \frac{x^3$ - 4x^2$ + 3x}{4x^2$ + 4x}$$

Question 16

For $$f(x) = \frac{x-1}{4x-8}$$, what is the vertical asymptote?

Accepted Answer

$$x = 2$$

Question 17

A rational function $$f(x) = \frac{x^3$ + x^2$ - 6x}{4x^3$ - 4x^2$ - 8x}$$ has holes at which values?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 18

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

Accepted Answer

$$x = -1$$ and $$x = 3$$

Question 19

For $$f(x) = \frac{1}{-3x^2$ + 9x}$$, what values are excluded from the domain?

Accepted Answer

$$x = 0$$ and $$x = 3$$

Step-by-Step Guidance for Rational Functions in College Algebra (Precalculus)

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