Q: 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?

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

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

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

Q: 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 $t = 3$?

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

Question 1

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

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 $t = 0$?

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: $C(x) = \begin{cases} 5 & 	ext{if } 0 \leq x \leq 2 \ 5 + 2(x-2) & 	ext{if } x > 2 \end{cases}$, 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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