Q: A biologist models the population of a bacteria culture with the function $P(t) = 500e^{0.3t}$, where $t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

$e^{1.5} \approx 4.48$, so $P(5) \approx 500 \times 4.48 = 2240$

Q: A company’s profit $P$ (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 $x = 2$ thousand units

Q: A car depreciates in value according to the function $V(t) = 20000(0.85)^t$, where $t$ is the number of years after purchase. What is the value of the car after 3 years?

$V(3) = 20000 \times (0.85)^3 \approx 12262.5$

Q: A medication is administered so that its concentration in the bloodstream decreases by 20% every hour. If the initial concentration is 50 mg/L, what will the concentration be after 3 hours?

$50 \times (0.8)^3 = 25.6$ mg/L

Question 1

Which of the following best describes the domain of the function $f(x) = \dfrac{1}{x^2 - 4}$?

Accepted Answer

$x 
eq 2$ and $x 
eq -2$

Question 2

A biologist models the population of a bacteria culture with the function $P(t) = 500e^{0.3t}$, where $t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

Accepted Answer

$e^{1.5} \approx 4.48$, so $P(5) \approx 500 	imes 4.48 = 2240$

Question 3

Which of the following is the inverse of the function $f(x) = 2x + 7$?

Accepted Answer

$f^{-1}(x) = \dfrac{x - 7}{2}$

Question 4

Solve the system of equations using any method:

$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$

Accepted Answer

$x = 2, y = 1$

Question 5

Which of the following is the correct factorization of the polynomial $x^3 - 4x$?

Accepted Answer

$x(x^2 - 4) = x(x - 2)(x + 2)$

Question 6

A radioactive substance decays according to $A(t) = 1000e^{-0.05t}$, where $t$ is in years. How long will it take for the substance to decay to 500 grams?

Accepted Answer

$t = \dfrac{\ln(0.5)}{-0.05} \approx 13.86$ years

Question 7

Which of the following is the solution set for the inequality $|2x - 3| < 5$?

Accepted Answer

$-1 < x < 4$

Question 8

Given $f(x) = x^2$ and $g(x) = x + 1$, what is $(f \circ g)(2)$?

Accepted Answer

$9$

Question 9

Which of the following is the correct solution to the equation $\log_2(x) = 5$?

Accepted Answer

$x = 32$

Question 10

A company’s profit $P$ (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 $x = 2$ thousand units

Question 11

Which of the following is the solution to the equation $2^{x+1} = 16$?

Accepted Answer

$x = 3$

Question 12

If $f(x) = \dfrac{3x - 1}{x + 2}$, what is the horizontal asymptote of $f(x)$?

Accepted Answer

$y = 3$

Question 13

Which of the following is the solution to the quadratic equation $x^2 - 6x + 5 = 0$?

Accepted Answer

$x = 1$ and $x = 5$

Question 14

A car depreciates in value according to the function $V(t) = 20000(0.85)^t$, where $t$ is the number of years after purchase. What is the value of the car after 3 years?

Accepted Answer

$V(3) = 20000 	imes (0.85)^3 \approx 12262.5$

Question 15

Which of the following is the solution to the equation $\ln(x) = 2$?

Accepted Answer

$x = e^2$

Question 16

Given the system $\begin{cases} x + y = 4 \ x - y = 2 \end{cases}$, what is the value of $x$?

Accepted Answer

$x = 3$

Question 17

Which of the following is the correct solution to the equation $|x + 2| = 7$?

Accepted Answer

$x = 5$ or $x = -9$

Question 18

If $f(x) = x^2 - 4x + 3$, what is the vertex of the parabola represented by $f(x)$?

Accepted Answer

$(2, -1)$

Question 19

Which of the following is the correct solution to the equation $\log_{10}(x) = -2$?

Accepted Answer

$x = 0.01$

Question 20

A medication is administered so that its concentration in the bloodstream decreases by 20% every hour. If the initial concentration is 50 mg/L, what will the concentration be after 3 hours?

Accepted Answer

$50 	imes (0.8)^3 = 25.6$ mg/L

College Algebra: Functions, Equations, and Applications – Study Guide

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