Q: Given the rational function $$f(x) = \frac{5x^2$ + 1}{2x - 6}$$, which of the following statements about its limits is correct?

$$\lim_{x \to 3^+} f(x) = -\infty$$

Q: Which of the following is true about the end behavior of $$f(x) = \frac{3x^2$ - 2x + 1}{5x^2$ - 3x + 4}$$?

$$\lim_{x \to \infty} f(x) = \frac{3}{5}$$

Question 1

Given the rational function $$f(x) = \frac{5x^2$ + 1}{2x - 6}$$, which of the following statements about its limits is correct?

Accepted Answer

$$\lim_{x 	o 3^+} f(x) = -\infty$$

Question 2

Which of the following is the correct equation for the horizontal asymptote of $$f(x) = \frac{3x^2$ - 5x^3$ + 1}{4x^2$ + 5x^4$ - 2x}$$?

Accepted Answer

$$y = 0$$

Question 3

The function $$f(x) = \frac{(5-x)(2x+1)^2$}{(3x-1)^2$(x+1)}$$ has which horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 4

For $$g(x) = \frac{x^2$ - 5x + 6}{x^2$ + 2x - 8}$$, what are the equations for the vertical asymptotes?

Accepted Answer

$$x = -4$$ and $$x = 2$$

Question 5

The function $$h(x) = \frac{(x+2)^2$(x-4)^3$$}{$$x(x+2)(x-4)^4$$}$ has vertical asymptotes at which values?

Accepted Answer

$$x = -2, x = 0, x = 4$$

Question 6

Which of the following is true about the end behavior of $$f(x) = \frac{3x^2$ - 2x + 1}{5x^2$ - 3x + 4}$$?

Accepted Answer

$$\lim_{x 	o \infty} f(x) = \frac{3}{5}$$

Question 7

The function $$f(x) = \frac{x^2$(x-4)}{(x-4)(x-2)}$$ has a hole at which $$x-$$value, and what is the value of the function at that hole?

Accepted Answer

$$\lim_{x 	o 4} f(x) = 8$$

Question 8

The function $$g(x) = \frac{2x^2$ - x - 3}{x^2$ - 1}$$ has a hole at what value(s) of $$x$$?

Accepted Answer

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

Question 9

For $$f(x) = \frac{(x-2)^2$(x-3)}{(x-2)(x-3)^2$}$$, at which coordinate is there a hole?

Accepted Answer

$$(2, -1)$$

Question 10

Given $$g(x) = \frac{(x-3)^2$(4-x)}{x^2$(x+4)}$$, at what values of $$x$$ are the zeros of $$g$$?

Accepted Answer

$$x = 3$$ and $$x = 4$$

Question 11

A rational function $$f(x) = \frac{x^2$ + 2x - 8}{x^2$ - x - 2} is $$given. Which of the following is the correct factorization of the numerator and denominator?

Accepted Answer

Numerator: $$(x+4)(x-2)$$, Denominator: $$(x-2)(x+1)$$

Question 12

Suppose $$f(x) = \frac{x^2$ - 2x - 15}{x^2$ + 2x - 3}. $$What are the $$x-$$intercepts of $$f$$?

Accepted Answer

$$x = 5$$ and $$x = -3$$

Question 13

A rational function $$f(x) = \frac{x^2$ - 3x - 10}{x + 2} is $$given. What is the equation of the slant (oblique) asymptote?

Accepted Answer

$$y = x - 5$$

Question 14

Which of the following best describes the Rule of Four (GNAW) approach in analyzing rational functions?

Accepted Answer

It requires students to use graphic, numeric, algebraic, and verbal representations.

Question 15

A rational function $$f(x)$$ has a degree of $$n in $$the numerator and $$m in $$the denominator. For very large $$x$$, the graph of $$f$$ approaches the line $$y = 3x - 5. $$What must be true about $$n$$ and $$m$$?

Accepted Answer

$$n - m = 1$$

Question 16

A rational function $$g(x)$$ has $$x-3 as a $$factor of both its numerator and denominator. What happens on the graph of $$g at$$ $$x=3$$?

Accepted Answer

The graph of $$g$$ has a hole at $$x=3.$$

Question 17

Suppose $$f(x) = \frac{x^2$ - 3x - 10}{2(x^2$ - 4x - 5)}. $$What are the vertical asymptotes of $$f$$?

Accepted Answer

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

Question 18

A rational function $$f(x) = \frac{x}{x^2$ + 1} is $$given. Which of the following statements is true about its asymptotes?

Accepted Answer

It has a horizontal asymptote at $$y = 0.$$

Rational Functions: Analysis, Graphs, and Applications in Precalculus

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