Q: A company’s profit $$P(x) in $$thousands of dollars is modeled by the function $$P(x) = 2x^3$ - 5x^2$ + 3x - 7$$, where $$x is $$the number of years since the company was founded. What type of function is this, and what is its end behavior as $$x \to \infty$$?

It is a cubic (power) function; as $$x \to \infty$$, $$P(x) \to \infty.$$

Q: If $$f(x) = x^4$ - 3x^2$ + 2$$, which of the following best describes its end behavior?

As $$x \to \infty$$, $$f(x) \to \infty$$; as $$x \to -\infty$$, $$f(x) \to \infty$$

Question 1

A company’s profit $$P(x) in $$thousands of dollars is modeled by the function $$P(x) = 2x^3$ - 5x^2$ + 3x - 7$$, where $$x is $$the number of years since the company was founded. What type of function is this, and what is its end behavior as $$x 	o \infty$$?

Accepted Answer

It is a cubic (power) function; as $$x 	o \infty$$, $$P(x) 	o \infty.$$

Question 2

Given $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is the domain of the combined function h(x) = f(g(x$$))$$?

Accepted Answer

$$[0, \infty)$$

Question 3

A rational function $$R(x) = \frac{2x^2$ + 3x - 5}{x^2$ - 4} is $$given. Which of the following describes its horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 4

Suppose $$f(x) = x^3$ and g(x) = 2x + 1$. What is the composition f(g(x$$))$$?

Accepted Answer

$$(2x + 1)^3$

Question 5

A population of bacteria doubles every 4 hours. Which function best models the population $$P(t)$$ after $$t$$ hours, if the initial population is 500?

Accepted Answer

$$P(t) = 500 \cdot 2$$^{t/4}$$

Question 6

Which of the following is the correct period of the function $$f(x) = 3 \sin(2x)$$?

Accepted Answer

$$\pi$$

Question 7

Given $$f(x) = \sin(x)$$, which transformation results in $$g(x) = 2\sin(x - \frac{\pi}{4}) + 1$$?

Accepted Answer

Vertical stretch by 2, right shift by $$\frac{\pi}{4}$$, up shift by 1

Question 8

If $$	an(	heta) = 3$$ and $$	heta is in $$the second quadrant, what is $$\sin(	heta)$$?

Accepted Answer

Question 9

A Ferris wheel rotates such that the height $$h(t) in $$meters of a seat above the ground at time $$t$$ seconds is given by $$h(t) = 10 \sin(\frac{\pi}{30} t) + 12. $$What is the amplitude and the midline of this function?

Accepted Answer

Amplitude: 10, Midline: $$y = 12$$

Question 10

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

Accepted Answer

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

Question 11

Given $$\sin$$^{-1}$$(\frac{1}{2})$$, what is the principal value?

Accepted Answer

$$\frac{\pi}{6}$$

Question 12

A function $$f(x)$$ has the property $$f(x) = f(x + 8). $$What is the period of $$f(x)$$?

Accepted Answer

$$8$$

Question 13

If $$f(x) = \cos(x)$$ and $$g(x) = 2f(x - \pi) - 1$$, which transformation is applied to $$f(x) to $$obtain $$g(x)$$?

Accepted Answer

Horizontal shift right by $$\pi$$, vertical stretch by 2, down shift by 1

Question 14

A triangle has side lengths $$a = 5$$, $$b = 7$$, and $$c = 10. $$Which law would you use to find the angle opposite side $$c$$?

Accepted Answer

Law of Cosines

Question 15

If $$f(x) = x^4$ - 3x^2$ + 2$$, which of the following best describes its end behavior?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty$$

Question 16

Which of the following is the correct expression for the sum of two sine functions: $$\sin(x) + \sin(2x)$$?

Accepted Answer

$$2\sin(\frac{3x}{2})\cos(\frac{x}{2})$$

Question 17

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

Accepted Answer

$$x = 2$$

Question 18

Given $$\cos(	heta) = -\frac{1}{2}$$ and $$	heta is in $$the third quadrant, what is $$	heta in $$radians?

Accepted Answer

$$\frac{7\pi}{6}$$

Precalculus Study Guide: Power, Rational, Periodic, and Trigonometric Functions

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