Question 1

Which of the following systems of equations has exactly one solution, and what is that solution?

$$\begin{cases} y = 4x - 3 \ 2y + 10x = 48 \end{cases}$$

Accepted Answer

Exactly one solution: $$x = 3$$, $$y = 9$$

Question 2

Given the relation $$\{(4, 9), (2, 7), (4, -5), (5, -7)\}$$, what is the domain and is this relation a function?

Accepted Answer

Domain: $$\{2, 4, 5\}$$; No, it is not a function because $$x=4$$ maps to two different $$y$$ values.

Question 3

Which of the following equations defines $$y as a $$function of $$x$$?

Accepted Answer

$$y = x^3$ - 64$$

Question 4

Find the average rate of change of $$f(x) = \sqrt{x}$$ from $$x_1 = 2$ to x_2$ = 8.$$

Accepted Answer

$$\dfrac{\sqrt{8} - \sqrt{2}}{8 - 2}$$

Question 5

Given the function $$f(x) = 5x + 1$$, what is its inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x - 1}{5}$$

Question 6

What is the standard form of the equation of a circle with center $$(-2, 9)$$ and radius $$10$$?

Accepted Answer

$$(x + 2)^2$ + (y - 9)^2$ = 100$$

Question 7

Given $$f(x) = x^2$ + 5x$$ from $$x_1 = 4$ to x_2$ = 6$$, what is the average rate of change?

Accepted Answer

$$\dfrac{f(6) - f(4)}{6 - 4}$$

Question 8

Which of the following describes the transformation needed to graph $$g(x) = |x - 2| + 4$$ from $$f(x) = |x|$$?

Accepted Answer

Shift right 2 units, shift up 4 units.

Question 9

Find the zeros of $$f(x) = 3x^4$ + 4x - x^3$ and state the multiplicity of each zero.

Accepted Answer

Question 10

Which of the following is the correct condensed form of $$\log_3$ 15 - \log_3$ 5$$?

Accepted Answer

$$\log_3$ 3$$

Question 11

If $$A = P\left(1 + \dfrac{r}{n}\$$right)^{nt}$$, what is the accumulated value of an investment of $$\$$1,500$$ for $$2$$ years at an interest rate of $$8\%$ compounded quarterly?

Accepted Answer

$$A = 1500\left(1 + \dfrac{0.08}{4}\$$right)^{8}$$

Question 12

Which of the following is the correct logarithmic form of $$5^3 = 125$?

Accepted Answer

$$\log_5$ 125 = 3$$

Question 13

Given $$f(x) = \dfrac{2x}{x^2$ - 9}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 14

Solve the equation $$2^{(x - 7)} = 4$ for x$.

Accepted Answer

$$x = 9$$

Question 15

What is the domain of $$f(x) = \sqrt{x - 5}$$?

Accepted Answer

$$x \geq 5$$

Question 16

Which of the following is the correct expansion of $$\log_2$ \left(\dfrac{\sqrt{8}}{5}\right)$$ using properties of logarithms?

Accepted Answer

$$\dfrac{1}{2} \log_2$ 8 - \log_2$ 5$$

Question 17

Given the quadratic function $$f(x) = x^2$ + 2x - 8$$, what is the axis of symmetry?

Accepted Answer

$$x = -2$$

Question 18

If $$f(x) = 3x^3$ + 2x^2$ - 5x - 6$$, use the Rational Zero Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm2, \pm3, \pm6, \pm\dfrac{1}{3}, \pm\dfrac{2}{3}, \pm\dfrac{3}{3}, \pm\dfrac{6}{3}$$

Question 19

Which of the following is the correct exponential form of $$\log_4$ 256 = x$$?

Accepted Answer

$$4^x = 256$$

Precalculus Final Exam Review: Key Concepts and Practice Problems

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