Q: A car's value depreciates exponentially according to $V(t) = 20,000 \times 0.85^t$, where $t$ is the number of years after purchase. What is the value after $3$ years?

$V(3) = 20,000 \times 0.85^3 \approx 12,281$

Question 1

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

Accepted Answer

A function that can be written as $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $n$ is a non-negative integer and $a_n, a_{n-1}, \ldots, a_0$ are real numbers.

Question 2

A business models its profit $P$ (in dollars) as a function of the number of units sold $x$ by $P(x) = -2x^2 + 400x - 10,000$. What is the maximum profit, and how many units must be sold to achieve it?

Question 3

Which of the following is the correct inverse of the exponential function $f(x) = 3^x$?

Accepted Answer

$f^{-1}(x) = \log_3(x)$

Question 4

A scientist is studying the decay of a radioactive substance. The amount $A$ (in grams) after $t$ years is given by $A(t) = 50e^{-0.2t}$. How much of the substance remains after $5$ years?

Accepted Answer

$A(5) = 50e^{-1} \approx 18.4$ grams

Question 5

Which of the following equations represents a linear function?

Accepted Answer

$f(x) = 2x + 5$

Question 6

A population of bacteria doubles every 3 hours. If the initial population is 500, which function models the population $P$ after $t$ hours?

Accepted Answer

$P(t) = 500 	imes 2^{t/3}$

Question 7

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

Accepted Answer

$x = 32$

Question 8

A company finds that the cost $C$ to produce $x$ items is $C(x) = 1000 + 15x$. If the company sells each item for $25$, how many items must be sold to break even?

Accepted Answer

$100$ items

Question 9

Which of the following is the correct graphing transformation for $f(x) = x^2$ to $g(x) = (x-3)^2 + 4$?

Accepted Answer

Shift right by $3$ units and up by $4$ units.

Question 10

A medical researcher models the concentration $C$ of a drug in the bloodstream as $C(t) = 20e^{-0.5t}$, where $t$ is in hours. What is the half-life of the drug?

Accepted Answer

$t = \frac{\ln(2)}{0.5} \approx 1.39$ hours

Question 11

Which of the following is the correct solution to the quadratic equation $x^2 - 6x + 8 = 0$?

Accepted Answer

$x = 2$ and $x = 4$

Question 12

A car's value depreciates exponentially according to $V(t) = 20,000 	imes 0.85^t$, where $t$ is the number of years after purchase. What is the value after $3$ years?

Accepted Answer

$V(3) = 20,000 	imes 0.85^3 \approx 12,281$

Question 13

Which of the following is the correct domain of the function $f(x) = \frac{1}{x-2}$?

Accepted Answer

All real numbers except $x = 2$

Question 14

A chemical reaction follows the equation $C(t) = 100e^{-0.3t}$, where $C$ is the concentration in mg/L and $t$ is time in hours. What is the concentration after $2$ hours?

Accepted Answer

$C(2) = 100e^{-0.6} \approx 54.9$ mg/L

Question 15

Which of the following is the correct solution to the system of equations: $\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$?

Accepted Answer

$x = 2$, $y = 3$

Question 16

Which of the following is the correct range of the function $f(x) = x^2$?

Accepted Answer

$[0, \infty)$

Question 17

A student invests $\$1,000$ in an account that earns $6\%$ annual interest compounded monthly. What is the value after $2$ years? (Use $A = P(1 + \frac{r}{n})^{nt}$)

Accepted Answer

$A = 1000(1 + \frac{0.06}{12})^{24} \approx \$1,127$

Question 18

Which of the following is the correct solution to $2^{x} = 10$?

Accepted Answer

$x = \log_2(10) \approx 3.32$

Question 19

A city’s population is modeled by $P(t) = 80,000e^{0.02t}$, where $t$ is years since 2020. What will the population be in 2030?

Accepted Answer

$P(10) = 80,000e^{0.2} \approx 97,712$

College Algebra: Course Overview and Key Concepts

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