Question 1

Given the function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}$$, which of the following describes its domain?

Accepted Answer

$$(-\infty, -2) \cup (-2, 3) \cup (3, \infty)$$

Question 2

Which of the following is the correct set of vertical and horizontal asymptotes for $$f(x) = \frac{2x^2$ + 2x + 1}{x^2$ - x - 6}$$?

Accepted Answer

Vertical: $$x = -2, x = 3$$; Horizontal: $$y = 2$$

Question 3

Given $$f(x) = x^3$ - 4x^2$ + 7x - 6$$, which of the following is a possible set of real zeros?

Accepted Answer

$$x = 1, x = 2, x = 3$$

Question 4

Find the range of the function $$f(x) = \sqrt{x - 1}.$$

Accepted Answer

$$[0, \infty)$$

Question 5

Which of the following functions has an oblique asymptote? Select the correct function and its asymptote.

Accepted Answer

$$f(x) = \frac{x^2$ + x + 1}{x - 2}$$; Oblique asymptote: $$y = x + 3$$

Question 6

Given $$f(x) = \sqrt{x - 1}$$ and $$g(x) = \frac{2}{x}$$, what is the domain of $$f(g(x))$$?

Accepted Answer

$$(-\infty, 0) \cup (0, 1]$$

Question 7

If $$f(x) = x^2$ - 4x + 7$$ and $$g(x) = 2x - 3$$, find $$(g \circ f)(1).$$

Accepted Answer

$$7$$

Question 8

Which of the following intervals is the solution to the inequality $$x^2 + 2x - 4$$ \u003c 0$$?

Accepted Answer

$$(-2 - \sqrt{6}, -2 + \sqrt{6})$$

Question 9

Given the rational function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}$$, which $$x-$$value corresponds to a hole in the graph?

Accepted Answer

$$x = 2$$

Question 10

Given $$f(x) = x^3$ - 4x^2$ + 7x - 6$$, use the Intermediate Value Theorem to show that there is a zero in the interval $$[1, 2]. $$Which statement is correct?

Accepted Answer

$$f(1)  0$$, so there is a zero in $$[1, 2]$$

Question 11

Find the inverse of the function $$f(x) = \sqrt{x - 2}.$$

Accepted Answer

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

Question 12

Given the polynomial $$f(x) = x^4$ + 4x^3$ + x - 6$$, which of the following is a possible set of zeros?

Accepted Answer

$$x = -3, x = 1, x = 2, x = -1$$

Question 13

Which of the following is the correct interval notation for the solution to the inequality $$x^2 - 4x + 3$$ \leq 0$$?

Accepted Answer

$$[1, 3]$$

Question 14

Given $$f(x) = \frac{2x}{x+1}$$ and $$g(x) = \frac{1}{x}$$, what is the domain of $$(g \circ f)(x)$$?

Accepted Answer

$$(-\infty, -1) \cup (-1, 0) \cup (0, \infty)$$

Question 15

Which of the following statements about the function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6} is $$true?

Accepted Answer

It has a vertical asymptote at $$x = -2$$ and a hole at $$x = 2$$

Question 16

Given the table below, where $$f(x)$$ and $$g(x)$$ are defined for $$x = 1, 2, 3, 4$$:

$$x$$: 1, 2, 3, 4
$$f(x)$$: 6, 5, 10, 4
$$g(x)$$: 2, 3, 4, 6

What is the value of $$(f \circ g)(3)$$?

Accepted Answer

$$5$$

Question 17

Which of the following is the correct inverse of $$f(x) = \frac{x^2$ - 2}{2}$$ for $$x \geq \sqrt{2}$$?

Accepted Answer

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

Question 18

Given $$f(x) = x^2$ - 4x + 7$$ and $$g(x) = 2x - 3$$, find $$(f \circ g)(1).$$

Accepted Answer

$$0$$

Question 19

Which of the following is the correct solution to the rational inequality $$\frac{x^2$ - 4}{x^2$ - x - 6} \leq 0$$?

Accepted Answer

$$(-\infty, -2) \cup (2, 3)$$

Polynomial, Rational, and Composite Functions: Study Guide and Practice Problems

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