Question 1

Factor the following expression completely: $$x^3$$ + $$5x^2 + 3x + 15$.

Accepted Answer

Question 2

Solve for $$x$$: $$\dfrac{x+6}{3} = \dfrac{x+8}{8}.$$

Accepted Answer

$$x = -2$$

Question 3

Solve for $$x$$: $$9x^2 - 35x = 4$.

Accepted Answer

Question 4

Solve: $$8x^2 + 3 = 1$$,155$$.

Accepted Answer

$$x = 12, \; x = -12$$

Question 5

Solve the absolute value inequality and express the solution set in interval notation: $$|x-4| - 2 \leq 3.$$

Accepted Answer

$$[-1, 9]$$

Question 6

Given the piecewise function $$h(x) = \begin{cases} x^2$-8 & x 
eq 2 \ x-3 & x = 2 \end{cases}$$, evaluate $$h(2)$$ and $$h(4).$$

Accepted Answer

$$h(2) = -1, \; h(4) = 8$$

Question 7

Find and simplify the difference quotient $$\dfrac{f(x+h)-f(x)}{h}$$ for $$f(x) = \dfrac{1}{3x}$$, $$h 
eq 0.$$

Accepted Answer

$$-\dfrac{1}{3x(x+h)}$$

Question 8

Write the equation for the line passing through $$(4,5)$$ and $$(5,2) in $$point-slope form.

Accepted Answer

$$y - 5 = -3(x - 4)$$

Question 9

Given $$f(x) = 3x^2$ - 9$$ and $$g(x) = 2x^3$ + 8$$, find $$(f+g)(x).$$

Accepted Answer

$$3x^2$$ + $$2x^3 - 1$

Question 10

Given $$f(x) = \dfrac{x-2}{5}$$ and $$g(x) = 5x + 2$$, find $$(g \circ f)(x)$$ and state the domain.

Accepted Answer

$$(g \circ f)(x) = x-2$$, domain: $$(-\infty, \infty)$$

Question 11

Find functions $$f$$ and $$g so $$that $$h(x) = (f \circ g)(x)$$, where $$h(x) = \dfrac{7}{\sqrt{6x+8}}.$$

Accepted Answer

$$f(x) = \dfrac{7}{\sqrt{x}}, \; g(x) = 6x+8$$

Question 12

Which two functions are inverses of each other for $$x \u003e 0$$? $$f(x) = \dfrac{x+2}{3}$$, $$g(x) = 3x+2$$, $$h(x) = \dfrac{x-2}{3}.$$

Accepted Answer

$$f$$ and $$g$$

Question 13

Find the inverse of the one-to-one function $$f(x) = 4x + 8.$$

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x-8}{4}$$

Question 14

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = 3x^2$ - 9$$, where $$x is $$the number of units sold (in hundreds). What is the profit if 5 units (i.e., 500 units) are sold?

Accepted Answer

$$66$$

Question 15

A medication’s concentration in the bloodstream is modeled by $$C(t) = \dfrac{1}{3t}$$, where $$t is $$time in hours. What is the average rate of change of concentration from $$t=2 to$$ $$t=2+h$$?

Accepted Answer

$$-\dfrac{1}{3 \cdot 2(2+h)}$$

Question 16

A line passes through the points $$(4,5)$$ and $$(5,2). $$What is its slope-intercept form?

Accepted Answer

$$y = -3x + 17$$

Question 17

Given $$f(x) = 3x^2$ - 9$$ and $$g(x) = 2x^3$ + 8$$, find $$(f \cdot g)(x).$$

Accepted Answer

$$6x^5$$ + $$24x^2$$ - $$18x^3 - 72$

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