Q: A piecewise function is defined as $f(x) = \begin{cases} x+2 & \text{if } x < 0 \\ 2x-1 & \text{if } x \geq 0 \end{cases}$. What is $f(-3) + f(2)$?

$(-3+2) + (2 \times 2 - 1) = -1 + 3 = 2$

Q: A chemical reaction’s rate $R$ (in mol/L·s) is modeled by $R(t) = 5t^2 - 20t + 25$. At what time $t$ is the reaction rate minimized, and what is the minimum rate?

Minimum at $t = 2$, $R(2) = 5(4) - 20(2) + 25 = 20 - 40 + 25 = 5$ mol/L·s

Q: A city’s population is modeled by $P(t) = 200,000(1.03)^t$, where $t$ is the number of years since 2020. What will the population be in 5 years (in 2025)? (Round to the nearest whole number.)

$P(5) = 200,000(1.03)^5 \approx 231,855$

Q: A medical test result is modeled by the function $T(x) = \frac{100}{1 + e^{-0.5x}}$. What is the value of $T(0)$?

$T(0) = \frac{100}{1 + 1} = 50$

Question 1

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

Accepted Answer

A relation where each input is paired with exactly one output.

Question 2

A pharmaceutical company models the concentration $C$ (in mg/L) of a drug in a patient's bloodstream $t$ hours after administration by $C(t) = 50e^{-0.2t}$. What is the concentration after 5 hours? (Round to two decimal places.)

Accepted Answer

$C(5) = 50e^{-1} \approx 18.39$ mg/L

Question 3

Which of the following is the domain of the function $f(x) = \sqrt{2x - 8}$?

Accepted Answer

$[4, \infty)$

Question 4

A company’s profit $P$ (in thousands of dollars) from selling $x$ units is modeled by $P(x) = -2x^2 + 40x - 100$. What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $P(10) = 100$ thousand dollars at $x = 10$ units.

Question 5

Which of the following is a one-to-one function?

Accepted Answer

$f(x) = 2x + 3$

Question 6

Solve the equation $3x^2 - 12x + 9 = 0$ for $x$.

Accepted Answer

$x = 1$ or $x = 3$

Question 7

A piecewise function is defined as $f(x) = \begin{cases} x+2 & 	ext{if } x < 0 \ 2x-1 & 	ext{if } x \geq 0 \end{cases}$. What is $f(-3) + f(2)$?

Accepted Answer

$(-3+2) + (2 	imes 2 - 1) = -1 + 3 = 2$

Question 8

Which of the following expressions is equivalent to $\frac{2x^2 - 8}{x^2 - 4x}$ for $x 
eq 0, 4$?

Accepted Answer

$\frac{2(x-2)}{x}$

Question 9

A biologist models the population of bacteria in a culture as $P(t) = 1000 \cdot 2^{t}$, where $t$ is the number of hours after the experiment begins. How many bacteria will there be after 3 hours?

Accepted Answer

$P(3) = 1000 \cdot 2^3 = 8000$

Question 10

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

Accepted Answer

$(2, 3)$

Question 11

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

Accepted Answer

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

Question 12

A chemical reaction’s rate $R$ (in mol/L·s) is modeled by $R(t) = 5t^2 - 20t + 25$. At what time $t$ is the reaction rate minimized, and what is the minimum rate?

Accepted Answer

Minimum at $t = 2$, $R(2) = 5(4) - 20(2) + 25 = 20 - 40 + 25 = 5$ mol/L·s

Question 13

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

Accepted Answer

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

Question 14

A student is solving the equation $\log_2(x) = 5$. What is the value of $x$?

Accepted Answer

$x = 32$

Question 15

Which of the following is the solution to the equation $\frac{1}{x} + \frac{1}{x-2} = 1$?

Accepted Answer

$x = 1$ or $x = 3$

Question 16

A city’s population is modeled by $P(t) = 200,000(1.03)^t$, where $t$ is the number of years since 2020. What will the population be in 5 years (in 2025)? (Round to the nearest whole number.)

Accepted Answer

$P(5) = 200,000(1.03)^5 \approx 231,855$

Question 17

Which of the following is the solution set for $|2x - 5| \leq 7$?

Accepted Answer

$-1 \leq x \leq 6$

Question 18

Which of the following is the correct simplification of $\frac{(x^2 - 9)}{(x^2 - 6x + 9)}$ for $x 
eq 3$?

Question 19

A medical test result is modeled by the function $T(x) = \frac{100}{1 + e^{-0.5x}}$. What is the value of $T(0)$?

Accepted Answer

$T(0) = \frac{100}{1 + 1} = 50$

MAT 115: College Algebra – Course Overview and Success Guide

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