Q: Which of the following is the correct limit statement for the vertical asymptote of $$G(x) = \frac{x^3$ - 5x^2$ + 6x}{x^2$ - 4} at$$ $$x = 2$$?

$$\lim_{x \to 2^-} G(x) = -\infty$$ and $$\lim_{x \to 2^+} G(x) = +\infty$$

Q: Which of the following limit statements corresponds to 'As the input values decrease without bound, the output values increase without bound'?

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

Question 1

Given the rational function $$F(x) = \frac{(2x-3)(x+4)}{(x-1)(x+4)}$$, which of the following statements is true about its domain?

Accepted Answer

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

Question 2

For the function $$F(x) = \frac{(2x-3)(x+4)}{(x-1)(x+4)}$$, what feature occurs at $$x = -4$$?

Accepted Answer

A hole

Question 3

Which of the following is the correct limit statement for the vertical asymptote of $$G(x) = \frac{x^3$ - 5x^2$ + 6x}{x^2$ - 4} at$$ $$x = 2$$?

Accepted Answer

$$\lim_{x 	o 2^-} G(x) = -\infty$$ and $$\lim_{x 	o 2^+} G(x) = +\infty$$

Question 4

Which of the following describes the end behavior of $$g(x) = \frac{3x^2$ - 5x + 2}{x + 1}$$?

Accepted Answer

g has a slant asymptote at $$y = 3x - 8$$, because the degree of the numerator exceeds the degree of the denominator.

Question 5

A rational function $$R(x)$$ has three x-intercepts, which account for a total multiplicity of 4 zeros. The degree of the denominator is 5, and the graph shows an oblique asymptote. How many nonreal zeros must $$R$$ have?

Accepted Answer

2

Question 6

Which of the following rational functions has a vertical asymptote at $$x = 4$$?

Accepted Answer

$$f(x) = \frac{(x+1)^2$}{(x-4)^2$}$$

Question 7

Given $$g(x) = \frac{(x^2$ + 3x)(x^2$ - 3x - 10)}{(x+1)(x^2$ - 2x)}$$, for which input values is $$g(x) = 0$$?

Accepted Answer

$$x = 0, -3, 5, -2$$

Question 8

A cubic function is shifted down 7 units, left 3 units, and horizontally dilated by a factor of $$\frac{1}{2}. $$Which of the following is the correct equation for the transformed function?

Accepted Answer

$$f(x) = \left(2(x+3)\$$right)^3$$ - 7$$

Question 9

Which of the following statements best describes the end behavior of $$M(x) = 1 - \frac{3}{(2x+4)^2$} as$$ $$x 	o \infty$$?

Accepted Answer

$$M(x) 	o 1$$

Question 10

For the function $$f(x) = \frac{(x+3)(x-2)}{(x+3)^2$(x-2)}$$, what feature occurs at $$x = -3$$?

Accepted Answer

A hole

Question 11

Which of the following is the correct factored form of $$j(x) = 9x^3$ + 30x^2$ + 16x$$?

Accepted Answer

$$j(x) = x(9x^2$ + 30x + 16)$$

Question 12

Which of the following limit statements corresponds to 'As the input values decrease without bound, the output values increase without bound'?

Accepted Answer

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

Question 13

Given $$p(x) = \frac{2x^2$ - 3x + 1}{x^2$ - 4x}$$, for which values of $$x$$ does $$p(x) = 4$$?

Accepted Answer

$$x = \frac{13 \pm \sqrt{187}}{4}$$

Question 14

Which of the following is the correct standard form of $$f(x) = (3x + 4)(x - 2)^3$?

Accepted Answer

$$f(x) = 3x^4$ - 20x^3$ + 43x^2$ - 24x - 64$$

Question 15

Which of the following describes the transformation for $$g(x) = -3|x + 4|$$?

Accepted Answer

Reflected about the x-axis, vertically stretched by 3, shifted left 4 units

Question 16

Given $$h(x) = \frac{x^3$ - x^2$}{x^2$ + 2x + 1}$$, what is the vertical asymptote?

Accepted Answer

$$x = -1$$

Question 17

Which of the following is the correct standard form of $$g(x) = (2x - 3)^5$?

Accepted Answer

$$g(x) = 32x^5$ - 240x^4$ + 720x^3$ - 1080x^2$ + 810x - 243$$

Precalculus Rational Functions and Transformations Review – Guided Study

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