Q: A piecewise function is defined as $$f(x) = \begin{cases} x+2 & \text{if } x \u003c 0 \\ 2x-1 & \text{if } x \geq 0 \end{cases}. $$What is $$f(-3) + f(2)$$?

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

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 $$f(x) = \begin{cases} x+2 & 	ext{if } x \u003c 0 \ 2x-1 & 	ext{if } x \geq 0 \end{cases}. $$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 $$x^2 - 5x + 6$$ \u003c 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 $$x 
eq 3$$?

Accepted Answer

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

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

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