Q: Which of the following is the correct definition of an exponential function?

$$f(x) = a^x$$, where $$a \u003e 0$$ and $$a \neq 1$$

Q: Given the piecewise function $$f(x) = \begin{cases} x^2$ & x \u003c 0 \\ 2x + 1 & x \geq 0 \end{cases}$$, what is the value of $$f(-3) + f(2)$$?

$$(-3)^2 + (2$$ \times 2 + 1) = 9 + 5 = 14$$

Q: A population of bacteria doubles every 3 hours. If the initial population is 500, what is the population after 9 hours?

$$500 \times 2$$^{9/3}$$ = 500 \times 2^3$ = 500 \times 8 = 4000$$

Q: Which of the following is the correct end behavior of the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3$$?

$$f(x) \to -\infty as$$ $$x \to \infty$$ and $$f(x) \to -\infty as$$ $$x \to -\infty$$

Q: Given the arithmetic sequence where $$a_1 = 7$ and d = 3$, what is the 10th term?

$$a_{10} = 7 + (10-1) \times 3 = 7 + 27 = 34$$

Q: Which of the following is the correct sum for the geometric series $$2 + 6 + 18 + 54$$?

$$2 \left( \frac{3^4$ - 1}{3 - 1} \right) = 2 \left( \frac{81 - 1}{2} \right) = 2 \times 40 = 80$$

Q: Which of the following is the correct form of DeMoivre’s Theorem for raising a complex number to the $$nth $$power?

$$[r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta)$$

Question 1

Which of the following is the correct definition of an exponential function?

Accepted Answer

$$f(x) = a^x$$, where $$a \u003e 0$$ and $$a 
eq 1$$

Question 2

Given the piecewise function $$f(x) = \begin{cases} x^2$ & x \u003c 0 \ 2x + 1 & x \geq 0 \end{cases}$$, what is the value of $$f(-3) + f(2)$$?

Accepted Answer

$$(-3)^2 + (2$$ 	imes 2 + 1) = 9 + 5 = 14$$

Question 3

Which of the following statements is TRUE about the relationship between exponential and logarithmic functions?

Accepted Answer

They are inverse functions of each other.

Question 4

Solve for $$x$$: $$2^{x+1} = 16$.

Accepted Answer

$$x = 3$$

Question 5

Which of the following is the correct domain of the function $$f(x) = \log_2$(x-3)$$?

Accepted Answer

$$x \u003e 3$$

Question 6

A population of bacteria doubles every 3 hours. If the initial population is 500, what is the population after 9 hours?

Accepted Answer

$$500 	imes 2$$^{9/3}$$ = 500 	imes 2^3$ = 500 	imes 8 = 4000$$

Question 7

Which of the following is the correct end behavior of the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3$$?

Accepted Answer

$$f(x) 	o -\infty as$$ $$x 	o \infty$$ and $$f(x) 	o -\infty as$$ $$x 	o -\infty$$

Question 8

Which conic section is represented by the equation $$x^2$$ + $$y^2 = 25$?

Accepted Answer

Circle

Question 9

Given the arithmetic sequence where $$a_1 = 7$ and d = 3$, what is the 10th term?

Accepted Answer

$$a_{10} = 7 + (10-1) 	imes 3 = 7 + 27 = 34$$

Question 10

Expand $$(x + 2)^3$ using the Binomial Theorem.

Accepted Answer

$$x^3$$ + $$6x^2 + 12x + 8$

Question 11

Which of the following is the correct formula for the Law of Cosines?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2 - 2ab$$ \cos C$$

Question 12

If $$\vec{u} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle 1, -2 \rangle$$, what is $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\langle 4, 2 \rangle$$

Question 13

Convert the rectangular coordinates $$(3, 3) to $$polar coordinates.

Accepted Answer

$$(3\sqrt{2}, \frac{\pi}{4})$$

Question 14

Which of the following is the correct sum for the geometric series $$2 + 6 + 18 + 54$$?

Accepted Answer

$$2 \left( \frac{3^4$ - 1}{3 - 1} ight) = 2 \left( \frac{81 - 1}{2} ight) = 2 	imes 40 = 80$$

Question 15

Which of the following is the correct equation for a parabola with vertex at $$(0,0)$$ and focus at $$(0,2)$$?

Accepted Answer

$$x^2 = 8y$

Question 16

Solve for $$x in $$the equation $$\sin x = \frac{1}{2}$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 17

Which of the following is the correct form of DeMoivre’s Theorem for raising a complex number to the $$nth $$power?

Accepted Answer

$$[r(\cos 	heta + i \sin 	heta)]^n = r^n (\cos n	heta + i \sin n	heta)$$

Question 18

Which of the following is the correct rectangular equation for the parametric equations $$x = 2t, y = t^2$?

Accepted Answer

$$y = \left( \frac{x}{2} \$$right)^2$$

Question 19

If $$\cos(2	heta) = 2\cos^2$	heta - 1$$, what is $$\cos(2 	imes 30^\circ)$$?

Accepted Answer

$$\cos(60^\circ) = 2 \left( \cos^2$ 30^\circ ight) - 1 = 2 \left( \frac{\sqrt{3}}{2} \$$right)^2$$ - 1 = 2 	imes \frac{3}{4} - 1 = \frac{3}{2} - 1 = \frac{1}{2}$$

Precalculus Algebra and Trigonometry: Course Competencies and Study Guide

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