Question 1

Given the function $$f(x) = \begin{cases} 2x + 1 & 	ext{if } x \u003c 0 \ x^2$ & 	ext{if } x \geq 0 \end{cases}$$, what is the value of $$f(-3)$$?

Accepted Answer

$$7$$

Question 2

A company’s profit $$P(x) in $$thousands of dollars is modeled by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$18$$ thousand dollars at $$x = 4$$ thousand units.

Question 3

Given $$f(x) = x^2$ - 4x + 3$$, find the $$x-$$intercepts of the function.

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 4

If $$g(x) = \sqrt{5 - x}$$, what is the domain of $$g(x) in $$interval notation?

Accepted Answer

$$(-\infty, 5]$$

Question 5

Given $$f(x) = 3x - 2$$ and $$g(x) = x^2$, find (f$$ \circ g)(2)$$.

Accepted Answer

$$3 	imes 4 - 2 = 10$$

Question 6

A population of bacteria is modeled by $$P(t) = 100e$$^{0.05t}$$, where t$ is time in hours. What is the average rate of change of the population from t = 0$ to t = 10$?

Accepted Answer

$$\dfrac{100e$$^{0.5}$$ - 100}{10}$$

Question 7

Given the graph of $$h(x)$$, you observe that $$h(x)$$ increases on $$(-\infty, 2)$$ and decreases on $$(2, \infty). $$What is the nature of $$x = 2$$?

Accepted Answer

$$x = 2 is a $$local maximum.

Question 8

A piecewise function is defined as $$f(x) = \begin{cases} x + 2 & 	ext{if } x \u003c 1 \ 3x - 1 & 	ext{if } x \geq 1 \end{cases}. $$What is $$f(1)$$?

Accepted Answer

Question 9

If $$f(x) = x^2$ and g(x) = 2x + 1$, what is f(g(3$$))$$?

Accepted Answer

$$49$$

Question 10

A medication’s concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the instantaneous rate of change at t = 0$?

Accepted Answer

Question 11

Given $$f(x) = \dfrac{1}{x-2}$$, what is the domain of $$f(x)$$?

Accepted Answer

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

Question 12

If $$f(x) = 2x + 3$$ and $$g(x) = x - 1$$, find $$(f + g)(4).$$

Accepted Answer

$$12$$

Question 13

A car’s position is given by $$s(t) = 4t^2$ + 2t$$, where $$t is in $$seconds. What is the average velocity from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

$$\dfrac{s(3) - s(1)}{3 - 1} = \dfrac{42 - 6}{2} = 18$$

Question 14

Given $$f(x) = x^2$ - 6x + 8$$, find the intervals where $$f(x) is $$increasing.

Accepted Answer

$$(3, \infty)$$

Question 15

If $$f(x) = x^2$ and g(x) = 2x + 1$, find (g$$ \circ f)(2)$$.

Accepted Answer

$$9$$

Question 16

A water tank is being filled such that the volume $$V(t) in $$liters after $$t$$ minutes is $$V(t) = 50 + 8t. $$What is the average rate of change of volume from $$t = 5 to$$ $$t = 15$$?

Accepted Answer

$$8$$ liters per minute

Question 17

Given $$f(x) = \dfrac{x+1}{x-1}$$, for which value of $$x is$$ $$f(x)$$ undefined?

Accepted Answer

$$x = 1$$

Question 18

If $$f(x) = 2x$$ and $$g(x) = x^2$ + 1$$, find $$f(g(2)).$$

Accepted Answer

$$10$$

Question 19

A piecewise function models the cost $$C(x) of $$parking at a city lot: $$C(x) = \begin{cases} 5 & 	ext{if } 0 \leq x \leq 2 \ 5 + 2(x-2) & 	ext{if } x \u003e 2 \end{cases}$$, where $$x is $$hours parked. What is the cost for parking $$4$$ hours?

Accepted Answer

$$9$$

Question 20

Given $$f(x) = x^2$ - 4$$ and $$g(x) = x + 2$$, find $$(f + g)(-2).$$

Accepted Answer

$$-2$$

Functions and Their Properties in College Algebra

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