Question 1

A biologist is tracking the population of a bacteria culture, which doubles every 3 hours. If the initial population is 500, what will the population be after 12 hours? Use the exponential growth formula $$P(t) = P_0$ \cdot 2$$^{t/3}$$, where t$ is in hours.

Accepted Answer

$$P(12) = 500 \cdot 2$$^{4}$$ = 500 \cdot 16 = 8000$$

Question 2

Find the distance between the points $$A(2, -3)$$ and $$B(-4, 5)$$ using the distance formula.

Accepted Answer

$$\sqrt{(2 + 4)^2$ + (-3 - 5)^2$} = \sqrt{6^2$ + (-8)^2$} = \sqrt{36 + 64} = \sqrt{100} = 10$$

Question 3

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

Accepted Answer

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

Question 4

A company’s revenue $$R (in $$thousands of dollars) is modeled by $$R(x) = 5x^2$ + 20x + 100$$, where $$x is $$the number of units sold (in thousands). What is the revenue if 3,000 units are sold?

Accepted Answer

$$R(3) = 5 \cdot 3^2$ + 20 \cdot 3 + 100 = 5 \cdot 9 + 60 + 100 = 45 + 60 + 100 = 205$$

Question 5

Which of the following is the correct composition of the functions $$f(x) = x^2$ and g(x) = x + 1$ evaluated at x = 2$ for (f$$ \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(2) = f(g(2)) = f(3) = 3^2$ = 9$$

Question 6

Solve the quadratic equation $$x^2 - 5x + 6 = 0$.

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 7

A medical researcher models the decay of a drug in the bloodstream by $$C(t) = 50e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is time in hours. What is the concentration after 5 hours?

Accepted Answer

$$C(5) = 50e$$^{-1}$$ \approx 50 \cdot 0.3679 \approx 18.4 mg/L$$

Question 8

Which of the following is the correct solution to the inequality $$2x - 7 \u003c 5$$?

Accepted Answer

$$x \u003c 6$$

Question 9

Given the function $$f(x) = \frac{1}{x - 2}$$, what is the domain of $$f$$?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 2$$

Question 10

Expand the logarithmic expression $$\ln(2x^3$)$$ using logarithm properties.

Accepted Answer

$$\ln(2x^3$) = \ln(2) + 3\ln(x)$$

Question 11

A chemical reaction follows the equation $$A = A_0e$$^{-kt}$$, where A_0$ = 100 mg$$, $$k = 0.1$$, and $$t = 10$$ hours. What is the amount of substance $$A$$ left after 10 hours?

Accepted Answer

$$A = 100e$$^{-1}$$ \approx 100 \cdot 0.3679 \approx 36.8 mg$$

Question 12

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

Accepted Answer

$$x = 5, y = 2$$

Question 13

Which of the following is the correct equation of a line passing through the points $$(1, 2)$$ and $$(4, 8)$$?

Accepted Answer

$$y = 2x$$

Question 14

A radioactive isotope has a half-life of 6 hours. If the initial amount is 200 mg, how much remains after 18 hours? Use $$A = A_0$ \left(\frac{1}{2}\$$right)^{t/6}$$.

Accepted Answer

$$A = 200 \left(\frac{1}{2}\$$right)^{3}$$ = 200 \cdot \frac{1}{8} = 25 mg$$

Question 15

Which of the following is the correct solution for $$\log_{2}(x) = 5$$?

Accepted Answer

$$x = 32$$

Question 16

Find the range of the function $$f(x) = \sqrt{x - 4}.$$

Accepted Answer

$$[0, \infty)$$

Question 17

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

Accepted Answer

Maximum profit at $$x = 10$$, $$P(10) = -2 \cdot 100 + 40 \cdot 10 - 100 = -200 + 400 - 100 = 100$$

Question 18

Solve the system of equations for the intersection point: $$\begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases}$$

Accepted Answer

$$x = 1, y = 3$$

Question 19

Which of the following is the correct formula for the inverse of $$f(x) = \frac{2x - 5}{3}$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{3x + 5}{2}$$

Math 142 Final Exam Review: College Algebra Study Guide

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