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

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

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?

Accepted Answer

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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