Q: A car's value depreciates by 15% each year. If the car is currently worth $\$20,000$, what will its value be after 3 years?

$20,000 \times (0.85)^3 \approx \$12,282$

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

$(f \circ g)(2) = f(g(2)) = f(3) = 9$

Q: A hospital monitors the temperature of a patient using the function $T(t) = 98.6 + 0.5\sin(\frac{\pi t}{12})$, where $t$ is hours after midnight. What is the patient's temperature at 6 AM?

$T(6) = 98.6 + 0.5\sin(\frac{\pi \times 6}{12}) = 98.6 + 0.5\sin(\frac{\pi}{2}) = 99.1$

Q: A pharmaceutical company models the concentration $C$ of a drug in the bloodstream by $C(t) = 50e^{-0.2t}$, where $t$ is hours after administration. How long will it take for the concentration to drop to $10$ mg/L?

$t = \frac{\ln(5)}{0.2} \approx 8.05$ hours

Q: Which of the following is the correct domain of the function $f(x) = \frac{1}{x-4}$?

All real numbers except $x = 4$

Question 1

Which of the following is the correct definition of a rational function?

Accepted Answer

A function that can be written as $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) 
eq 0$.

Question 2

Given the function $f(x) = 2x^2 - 3x + 5$, what is the value of $f(-2)$?

Accepted Answer

$f(-2) = 19$

Question 3

A chemical reaction rate $R$ is modeled by the exponential function $R(t) = 100e^{-0.05t}$, where $t$ is time in minutes. What is the reaction rate after 20 minutes?

Accepted Answer

$R(20) = 100e^{-1} \approx 36.79$

Question 4

Which of the following is the inverse of the function $f(x) = 3x + 7$?

Accepted Answer

$f^{-1}(x) = \frac{x-7}{3}$

Question 5

A population of bacteria doubles every 4 hours. If the initial population is 500, which equation models the population $P$ after $t$ hours?

Accepted Answer

$P(t) = 500 \cdot 2^{t/4}$

Question 6

Solve the system of equations: $\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$

Question 7

Which of the following expressions is equivalent to $\log_{2}8$?

Accepted Answer

$3$

Question 8

A car's value depreciates by 15% each year. If the car is currently worth $\$20,000$, what will its value be after 3 years?

Accepted Answer

$20,000 	imes (0.85)^3 \approx \$12,282$

Question 9

Which of the following is the correct solution set for the inequality $x^2 - 5x + 6 < 0$?

Accepted Answer

$2 < x < 3$

Question 10

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

Accepted Answer

$(f \circ g)(2) = f(g(2)) = f(3) = 9$

Question 11

A hospital monitors the temperature of a patient using the function $T(t) = 98.6 + 0.5\sin(\frac{\pi t}{12})$, where $t$ is hours after midnight. What is the patient's temperature at 6 AM?

Accepted Answer

$T(6) = 98.6 + 0.5\sin(\frac{\pi 	imes 6}{12}) = 98.6 + 0.5\sin(\frac{\pi}{2}) = 99.1$

Question 12

Which of the following is the correct factorization of $x^3 - 8$?

Accepted Answer

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

Question 13

A pharmaceutical company models the concentration $C$ of a drug in the bloodstream by $C(t) = 50e^{-0.2t}$, where $t$ is hours after administration. How long will it take for the concentration to drop to $10$ mg/L?

Accepted Answer

$t = \frac{\ln(5)}{0.2} \approx 8.05$ hours

Question 14

Which of the following is the correct solution to the equation $2^{x} = 32$?

Accepted Answer

$x = 5$

Question 15

A quadratic function models the height $h$ (in meters) of a ball thrown upward: $h(t) = -4.9t^2 + 20t + 2$. What is the maximum height reached by the ball?

Accepted Answer

$h_{max} = 22.41$ meters

Question 16

Which of the following is the correct domain of the function $f(x) = \frac{1}{x-4}$?

Accepted Answer

All real numbers except $x = 4$

Question 17

A business's profit $P$ is modeled by $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of units sold. How many units should be sold to maximize profit?

Accepted Answer

$x = 10$

Question 18

Which of the following is the correct solution to $|3x - 7| = 5$?

Accepted Answer

$x = 4$ or $x = \frac{2}{3}$

College Algebra Syllabus and Course Structure Study Guide

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