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

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

Q: A hospital uses the matrix $$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} to $$represent patient flow between departments. What is the determinant of $$A$$?

$$2 \times 4 - 1 \times 3 = 5$$

Q: A company’s cost function is $$C(x) = 0.5x^2$ + 20x + 100. $$What is the average cost per item when $$x = 10$$?

Average cost is $$\frac{C(10)}{10} = \frac{0.5 \times 100 + 20 \times 10 + 100}{10} = \frac{5 + 200 + 100}{10} = \frac{305}{10} = 30.5$$

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, \ldots, a_0$ are real numbers.

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream over time with the function $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What type of function is this?

Accepted Answer

Exponential function

Question 3

Which of the following is the domain of the function $$f(x) = \frac{1}{x-3}$$?

Accepted Answer

All real numbers except $$x = 3$$

Question 4

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

Accepted Answer

Maximum profit is $$P(10) = 100$$ thousand dollars when $$x = 10$$

Question 5

Which of the following equations represents a circle with center $$(2, -3)$$ and radius $$5$$?

Accepted Answer

$$(x-2)^2$$ + $$(y+3)^2 = 25$

Question 6

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

Accepted Answer

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

Question 7

Which of the following is the solution set for $$|2x - 5| = 9$$?

Accepted Answer

$$x = 7 or$$ $$x = -2$$

Question 8

A car rental company charges a flat fee of $$\30$ plus $$\$$0.20$$ per mile driven. Which equation models the total cost $$C$$ for driving $$m$$ miles?

Accepted Answer

$$C = 30 + 0.20m$$

Question 9

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

Accepted Answer

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

Question 10

A biologist uses the function $$f(x) = \ln(x) to $$model the growth rate of a species. What is the domain of $$f(x)$$?

Accepted Answer

All real numbers $$x \u003e 0$$

Question 11

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

Accepted Answer

$$x = 2$$, $$y = 1$$

Question 12

Which of the following is a rational function?

Accepted Answer

$$f(x) = \frac{x^2$ + 1}{x - 4}$$

Question 13

A business’s revenue $$R is $$given by $$R(x) = 100x - x^2$, where x$ is the number of items sold. What is the maximum revenue?

Accepted Answer

$$R(50) = 2500$$

Question 14

Which of the following describes the range of the function $$f(x) = x^2$ - 4$$?

Accepted Answer

All real numbers $$y \geq -4$$

Question 15

A scientist models the decay of a radioactive substance with $$N(t) = N_0e$$^{-kt}$$. If N_0$ = 100$$ grams and $$k = 0.05$$, how much remains after $$10$$ years?

Accepted Answer

$$N(10) = 100e$$^{-0.5}$$ \approx 60.65$$ grams

Question 16

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

Accepted Answer

$$x = 2 or$$ $$x = 4$$

Question 17

A hospital uses the matrix $$A = \begin{bmatrix} 2 & 3 \ 1 & 4 \end{bmatrix} to $$represent patient flow between departments. What is the determinant of $$A$$?

Accepted Answer

$$2 	imes 4 - 1 	imes 3 = 5$$

Question 18

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

Accepted Answer

$$x = 32$$

Question 19

A company’s cost function is $$C(x) = 0.5x^2$ + 20x + 100. $$What is the average cost per item when $$x = 10$$?

Accepted Answer

Average cost is $$\frac{C(10)}{10} = \frac{0.5 	imes 100 + 20 	imes 10 + 100}{10} = \frac{5 + 200 + 100}{10} = \frac{305}{10} = 30.5$$

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