Q: Given the piecewise-defined 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: Which of the following is the sum of the first 10 terms of the arithmetic sequence where $$a_1 = 3$ and d = 5$?

$$S_{10} = \frac{10}{2}(2 \times 3 + 9 \times 5) = 5(6 + 45) = 5 \times 51 = 255$$

Question 1

Which of the following best describes the inverse relationship between exponential and logarithmic functions?

Accepted Answer

The logarithmic function undoes the effect of the exponential function.

Question 2

Given the piecewise-defined 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 is the correct domain for the function $$f(x) = \ln(x-2)$$?

Accepted Answer

$$x \u003e 2$$

Question 4

A population of bacteria doubles every 3 hours. If the initial population is 500, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P = 500 \cdot 2$$^{t/3}$$

Question 5

Which of the following is the correct end behavior for the polynomial $$f(x) = -2x^5$ + 3x^3$ - x + 7 as$$ $$x 	o \infty$$?

Accepted Answer

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

Question 6

Which of the following is the Fundamental Theorem of Algebra?

Accepted Answer

Every polynomial equation of degree $$n$$ with complex coefficients has exactly $$n$$ roots in the complex number system.

Question 7

Given the rational function $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}$$, which of the following describes all vertical asymptotes?

Accepted Answer

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

Question 8

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

Accepted Answer

$$x = 3$$

Question 9

Which of the following is the sum of the first 10 terms of the arithmetic sequence where $$a_1 = 3$ and d = 5$?

Accepted Answer

$$S_{10} = \frac{10}{2}(2 	imes 3 + 9 	imes 5) = 5(6 + 45) = 5 	imes 51 = 255$$

Question 10

Which of the following is the correct equation for an ellipse centered at the origin with a major axis of length 10 along the $$x-$$axis and a minor axis of length 6 along the $$y-$$axis?

Accepted Answer

$$\frac{x^2$}{25} + \frac{y^2$}{9} = 1$$

Question 11

If $$\sin(	heta) = \frac{3}{5}$$ and $$	heta is in $$the first quadrant, what is $$\cos(	heta)$$?

Accepted Answer

$$\frac{4}{5}$$

Question 12

Which of the following is the correct expansion of $$(x + 2)^3$ using the Binomial Theorem?

Accepted Answer

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

Question 13

Given the matrix $$A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$, what is $$\det(A)$$?

Accepted Answer

$$-2$$

Question 14

Which of the following is the correct solution to the system of equations: $$\begin{cases} x + y = 5 \ x - y = 1 \end{cases}$$?

Accepted Answer

$$x = 3, y = 2$$

Question 15

Which of the following is the correct polar form of the complex number $$-1 + i\sqrt{3}$$?

Accepted Answer

$$2\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)$$

Question 16

Which of the following is the correct sum for the geometric series $$2 + 6 + 18 + \ldots + 486$$?

Accepted Answer

$$728$$

Question 17

Which of the following is the correct solution to $$\log_2$(x) + \log_2$(x-2) = 3$$?

Accepted Answer

$$x = 4$$

Question 18

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 19

Which of the following is the correct formula for the sum of the first $$n$$ terms of a geometric sequence with first term $$a$$ and common ratio $$r$$ ($$r 
eq 1$$)?

Accepted Answer

$$S_n = a\frac{1 - r^n}{1 - r}$$

Precalculus Algebra and Trigonometry: Syllabus, Competencies, and Course Structure

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