Q: Solve the rational equation and state all restrictions: $$\displaystyle \frac{1}{x-2} + \frac{2}{x+2} = \frac{8}{x^2$-4}$$

Solution: $$x=4$$; Restrictions: $$x \neq 2, -2$$

Question 1

Which of the following is the correct simplification of the expression $$\displaystyle \frac{(-2x$$^{-3}$$y^2)^2$}{4x$$^{-2}$$y^{-4}$$}$? Write your answer with only positive exponents.

Accepted Answer

$$\displaystyle x^4$y^8$

Question 2

A function $$f(x) is $$defined as follows:

$$\displaystyle f(x) = \begin{cases} 3x+2, & x  0 \end{cases}$$

Which of the following statements is TRUE about the graph of $$f(x)$$?

Accepted Answer

There is an open circle at $$x=0 on $$the line $$y=-2x+1.$$

Question 3

Simplify the following expression and write your answer in standard form $$a+bi$$:

$$\displaystyle (2 - \sqrt{-16})^2$

Accepted Answer

$$\displaystyle 36 + 0i$$

Question 4

Solve the rational equation and state all restrictions:

$$\displaystyle \frac{1}{x-2} + \frac{2}{x+2} = \frac{8}{x^2$-4}$$

Accepted Answer

Solution: $$x=4$$; Restrictions: $$x 
eq 2, -2$$

Question 5

Given $$f(x) = 2x^2$ - 8x$$ and $$g(x) = x^2$ - 9$$, what is the domain of $$\displaystyle \frac{f}{g}(x) in $$interval notation?

Accepted Answer

$$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$$

Question 6

Consider the system of equations:

$$\displaystyle \begin{cases} y = 3x - 1 \ 6x - 2y = 8 \end{cases}$$

How many solutions does this system have?

Accepted Answer

Exactly one solution

Question 7

Find the domain of the function $$f(x) = \ln(2x - 10).$$

Accepted Answer

$$(5, \infty)$$

Question 8

Solve for $$x in $$the equation $$2^{x+1}$$ = $$5^{2x-3}. $$Give the exact answer in logarithmic form.

Accepted Answer

$$\displaystyle x = \frac{\ln 2 + 3\ln 5}{2\ln 5 - \ln 2}$$

Question 9

Let $$f(x) = x^2$ - 6x + 5$$ and $$g(x) = x - 2. $$Find $$(f \circ g)(3). $$

Accepted Answer

$$\displaystyle f(g(3)) = f(1) = 0$$

Question 10

Which of the following functions is NOT one-to-one on its entire domain?

Accepted Answer

$$h(x) = |x|$$

Question 11

Suppose $$f(x) = x^2$ - 4x + 1$$ with domain restricted to $$[2, \infty). $$What is $$f^{-1}(10$$)$$?

Accepted Answer

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

Question 12

Let $$f(x) = \frac{1}{x-3}$$ and $$g(x) = 6x. $$Find $$(f \circ g)(x)$$ and state its domain.

Accepted Answer

$$\displaystyle (f \circ g)(x) = \frac{1}{6x-3}$$, domain: $$x 
eq \frac{1}{2}$$

Question 13

Expand the logarithmic expression:

$$\displaystyle \log \left( \frac{4x^3$ \sqrt{y}}{(x+2)^2$} \right)$$

Accepted Answer

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

Question 14

A radioactive substance decays according to $$A = 1000e$$^{-0.02t}$$. How much remains after t = 50$ years? Round to the nearest whole number.

Accepted Answer

$$\displaystyle 368$$

Question 15

Solve for $$x$$ and round to four decimal places:

$$0.7^x = 5$$

Accepted Answer

$$\displaystyle x = \frac{\ln 5}{\ln 0.7} \approx -4.1206$$

Question 16

Which of the following best describes the relationship between the graphs of $$y = e^x$$ and $$y = \ln x$$?

Accepted Answer

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

Question 17

Rewrite the equation $$5x - 2y + 8 = 0 in $$slope-intercept form and identify the slope and y-intercept.

Accepted Answer

Slope: $$\displaystyle \frac{5}{2}$$, y-intercept: $$4$$

Question 18

A large rectangle has dimensions $$x+6 by$$ $$x+4. A $$smaller rectangle inside it has dimensions $$x+2 by$$ $$x. $$Write a polynomial in standard form for the shaded area between the two rectangles.

Accepted Answer

$$\displaystyle x^2$ + 8x + 24$$

Question 19

A quadratic model $$y = ax^2$ + bx + c$$ passes through the points $$(1, 10)$$, $$(2, 15)$$, and $$(3, 22). $$Which of the following is the correct system of equations for $$a$$, $$b$$, and $$c$$?

Accepted Answer

$$\begin{cases} a + b + c = 10 \ 4a + 2b + c = 15 \ 9a + 3b + c = 22 \end{cases}$$

Question 20

Which is the first nontrivial factoring method you would use for the expression $$12x^2 - 27$?

Accepted Answer

Greatest Common Factor (GCF)

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