Question 1

Given the rational function $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$, which of the following statements about its asymptotes is correct?

Accepted Answer

It has a horizontal asymptote at $$y = 2$$ and vertical asymptotes at $$x = -2$$ and $$x = 2.$$

Question 2

What is the $$y-$$intercept of the function $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$?

Accepted Answer

$$\dfrac{1}{2}$$

Question 3

For the function $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$, which of the following describes its domain?

Accepted Answer

$$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

Question 4

Which of the following best describes the symmetry of $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$?

Accepted Answer

It is an even function, symmetric with respect to the $$y-$$axis.

Question 5

What are the $$x-$$intercepts of $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$?

Accepted Answer

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

Question 6

A rational function has a numerator and denominator of equal degree. What determines the horizontal asymptote?

Accepted Answer

The ratio of the leading coefficients of the numerator and denominator.

Question 7

If the degree of the numerator of a rational function exceeds the degree of the denominator by 2, what is the end behavior asymptote?

Accepted Answer

A quadratic function.

Question 8

Which of the following intervals describes where $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4} is $$increasing?

Accepted Answer

$$(-\infty, -2)$$ and $$(2, 0)$$

Question 9

What is the local maximum value of $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$ and at what $$x$$ does it occur?

Accepted Answer

Local maximum of $$\dfrac{1}{2} at$$ $$x = 0$$

Question 10

A chemical reaction's rate is modeled by $$R(t) = \dfrac{2t^2$ - 2}{t^2$ - 4}$$, where $$t is $$time in seconds. What happens to the reaction rate as $$t$$ becomes very large?

Accepted Answer

It approaches $$2.$$

Question 11

Suppose a rational function $$f(x)$$ has vertical asymptotes at $$x = -2$$ and $$x = 2. $$Which of the following could be its denominator?

Accepted Answer

$$x^2 - 4$

Question 12

If $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$, what is $$\displaystyle \lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$2$$

Question 13

A rational function models the concentration of a drug in the bloodstream: $$C(t) = \dfrac{2t^2$ - 2}{t^2$ - 4}. $$For which values of $$t is $$the concentration undefined?

Accepted Answer

$$t = -2$$ and $$t = 2$$

Question 14

Which of the following best describes the range of $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$?

Accepted Answer

$$(-\infty, \dfrac{1}{2}] \cup (2, \infty)$$

Question 15

A rational function $$f(x)$$ has a horizontal asymptote at $$y = 2. $$Which of the following could be the leading coefficients of its numerator and denominator?

Accepted Answer

Numerator: $$2$$, Denominator: $$1$$

Question 16

If a rational function has a quadratic numerator and a quadratic denominator, what is the possible degree of the end behavior asymptote?

Accepted Answer

Zero (horizontal asymptote)

Question 17

Which of the following is true about the function $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$?

Accepted Answer

It is unbounded as $$x$$ approaches the vertical asymptotes.

Question 18

A rational function $$f(x)$$ has a horizontal asymptote at $$y = 0. $$Which of the following must be true?

Accepted Answer

The degree of the numerator is less than the degree of the denominator.

Question 19

Suppose $$f(x) = \dfrac{2x^2$ - 2}{x^2$ - 4}$$ models the height (in meters) of a projectile at time $$x (in $$seconds). For which values of $$x is $$the model not valid?

Accepted Answer

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

Rational Functions: Analysis, Graphs, and Asymptotes

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