Question 1

Which of the following correctly represents the domain of the function $$f(x) = \frac{5x^2$ - 10x}{x^2$ + 4x - 12} in $$interval notation?

Accepted Answer

$$(-\infty, -6) \cup (-6, 2) \cup (2, \infty)$$

Question 2

Simplify the expression $$\frac{(-3x$$^{-2}$$y^{-1})^2$}{(6x$$^{-3}$$y)^3$$}$ and write your answer with only positive exponents.

Accepted Answer

$$\frac{9x^4$}{216y^5$}$$

Question 3

A material decays according to $$A = 600e$$^{-0.004t}$$. How much remains after t = 120$ years? Round to the nearest pound.

Accepted Answer

$$A \approx 276$$ pounds

Question 4

Solve the equation $$0.8^x = 12$$ and round your answer to four decimal places.

Accepted Answer

$$x \approx -10.4261$$

Question 5

Which of the following is the first nontrivial factoring method for $$16x^2 - 81$?

Accepted Answer

Difference of Squares (DOS)

Question 6

Let $$f(x) = x^2$ - 4x + 1$$ and $$g(x) = x + 3. $$What is $$(f \circ g)(-1)$$?

Accepted Answer

$$(2)^2 - 4(2) + 1 = -3$

Question 7

Expand the logarithmic expression $$\log \left( \frac{5x^3$ \sqrt{x} - 1}{2(x + 4)^3$} \right)$$ using product, quotient, and power rules.

Accepted Answer

$$\log(5) + 3\log(x) + \frac{1}{2}\log(x) - \log(1) - \log(2) - 3\log(x + 4)$$

Question 8

Solve the quadratic equation $$2x^2 - 7x = 15$ and state the discriminant.

Accepted Answer

Discriminant $$D = 49 - 120 = -71$$, solutions are complex.

Question 9

Which of the following is the domain of $$f(x) = \ln\left( \frac{1}{x - 5} \right)$$?

Accepted Answer

$$(-\infty, 5)$$

Question 10

A large rectangle has dimensions $$x + 7 by$$ $$x + 5. A $$smaller rectangle inside it has dimensions $$x + 3 by$$ $$x + 1. $$What is the polynomial in standard form for the shaded area between the two rectangles?

Accepted Answer

$$(x + 7)(x + 5) - (x + 3)(x + 1) = x^2$ + 8x + 32$$

Question 11

Which of the following statements about the graphs of $$y = e^x$$ and $$y = \ln x is $$correct?

Accepted Answer

They are reflections of each other across the line $$y = x.$$

Question 12

Solve for $$x in $$the equation $$3^{x-2}$$ = $$7^{2x+1}$$ and give the exact answer in logarithmic form.

Accepted Answer

Question 13

Given the system $$\begin{cases} y = -2x + 5 \ 4x + 2y = 8 \end{cases}$$, how many solutions does the system have?

Accepted Answer

Exactly one solution

Question 14

Which of the following is true about the function $$k(x) = (x - 2)^2$?

Accepted Answer

It is not one-to-one on $$\mathbb{R}$$, but has an inverse if the domain is restricted to $$[2, \infty).$$

Question 15

Let $$f(x) = \frac{1}{x - 4}$$ and $$g(x) = \frac{12}{x}. $$What is $$(f \circ g)(x)$$ and its domain?

Accepted Answer

$$(f \circ g)(x) = \frac{1}{\frac{12}{x} - 4}$$, domain $$x 
eq 0, x 
eq 3$$

Question 16

Which value is excluded from the domain when performing the operation $$\frac{4x - 5}{x^2$ - 16} - \frac{x - 9}{x^2$ - 16}$$?

Accepted Answer

$$x = 4$$ and $$x = -4$$

Question 17

Which of the following is the slope and y-intercept of the line $$3x - 4y + 12 = 0$$?

Accepted Answer

Question 18

Which of the following is the correct standard form for the result of $$\left(4 - \sqrt{-49}\$$right)^2$$?

Accepted Answer

$$(4 - 7i)^2$ = 16 - 56i + 49i^2$ = -33 - 56i$$

Question 19

Suppose $$f(x) = x^2$ - 4x + 1$$ with domain $$[2, \infty). $$Estimate $$f^{-1}(10$$)$$.

Accepted Answer

$$f^{-1}(10) = 5$

Comprehensive Study Guide for College Algebra Final Exam

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