Q: A pharmaceutical company models the decay of a drug in the bloodstream with $$C(t) = C_0$ e$$^{-kt}$$, where C_0$ is the initial concentration and k$ is a positive constant. If C_0$ = 100 mg/L $$and $$k = 0.2$$, what is the concentration after 5 hours?

$$C(5) = 100 \times e$$^{-0.2 \times 5}$$ = 100 \times e$$^{-1}$$ \approx 36.79 mg/L$$

Q: A company’s revenue is modeled by $$R(x) = 50x - 0.5x^2$, where x$ is the number of units sold. What is the maximum revenue, and for how many units sold does it occur?

Maximum revenue is $$R(50) = 50 \times 50 - 0.5 \times 50^2$ = 2500 - 1250 = 1250 at$$ $$x = 50$$

Q: Solve the equation $$2x^2 - 5x + 3 = 0$ using the quadratic formula.

$$x = \frac{5 \pm \sqrt{(-5)^2$ - 4 \times 2 \times 3}}{2 \times 2} = \frac{5 \pm \sqrt{25 - 24}}{4} = \frac{5 \pm 1}{4}$$, so $$x = 1.5 or$$ $$x = 1$$

Q: A population of bacteria grows according to $$P(t) = 2000 e$$^{0.3t}$$. How many bacteria will there be after 4 hours?

$$P(4) = 2000 \times e$$^{0.3 \times 4}$$ = 2000 \times e$$^{1.2}$$ \approx 6642$$

Q: Which of the following describes the end behavior of the function $$f(x) = -2x^3$ + 5x^2$ - x + 7$$?

As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to \infty$$

Q: A car’s value depreciates according to $$V(t) = 20000 \times (0.85)^t$$, where $$t is $$the number of years. What is the value after 3 years?

$$V(3) = 20000 \times (0.85)^3$ = 20000 \times 0.614125 = 12282.50$$

Q: Which of the following is the correct composition of $$f(x) = x^2$ and g(x) = 2x + 1$ evaluated at x = 3$?

$$f(g(3)) = (2 \times 3 + 1)^2$ = (6 + 1)^2$ = 49$$

Question 1

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

Accepted Answer

A relation in which every input is paired with exactly one output.

Question 2

Given the function $$f(x) = 2x^2$ - 3x + 5$$, what is the domain of $$f$$?

Accepted Answer

All real numbers $$x$$

Question 3

A pharmaceutical company models the decay of a drug in the bloodstream with $$C(t) = C_0$ e$$^{-kt}$$, where C_0$ is the initial concentration and k$ is a positive constant. If C_0$ = 100 mg/L $$and $$k = 0.2$$, what is the concentration after 5 hours?

Accepted Answer

$$C(5) = 100 	imes e$$^{-0.2 	imes 5}$$ = 100 	imes e$$^{-1}$$ \approx 36.79 mg/L$$

Question 4

Which of the following equations represents a circle?

Accepted Answer

$$x^2$$ + $$y^2$$ = $$r^2$$

Question 5

A company’s revenue is modeled by $$R(x) = 50x - 0.5x^2$, where x$ is the number of units sold. What is the maximum revenue, and for how many units sold does it occur?

Accepted Answer

Maximum revenue is $$R(50) = 50 	imes 50 - 0.5 	imes 50^2$ = 2500 - 1250 = 1250 at$$ $$x = 50$$

Question 6

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

Accepted Answer

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

Question 7

Solve the equation $$2x^2 - 5x + 3 = 0$ using the quadratic formula.

Accepted Answer

$$x = \frac{5 \pm \sqrt{(-5)^2$ - 4 	imes 2 	imes 3}}{2 	imes 2} = \frac{5 \pm \sqrt{25 - 24}}{4} = \frac{5 \pm 1}{4}$$, so $$x = 1.5 or$$ $$x = 1$$

Question 8

A population of bacteria grows according to $$P(t) = 2000 e$$^{0.3t}$$. How many bacteria will there be after 4 hours?

Accepted Answer

$$P(4) = 2000 	imes e$$^{0.3 	imes 4}$$ = 2000 	imes e$$^{1.2}$$ \approx 6642$$

Question 9

Which of the following describes the end behavior of the function $$f(x) = -2x^3$ + 5x^2$ - x + 7$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty$$

Question 10

Solve the inequality $$|2x - 3| \u003c 5.$$

Accepted Answer

$$-1 \u003c x \u003c 4$$

Question 11

A car’s value depreciates according to $$V(t) = 20000 	imes (0.85)^t$$, where $$t is $$the number of years. What is the value after 3 years?

Accepted Answer

$$V(3) = 20000 	imes (0.85)^3$ = 20000 	imes 0.614125 = 12282.50$$

Question 12

Which of the following is the correct composition of $$f(x) = x^2$ and g(x) = 2x + 1$ evaluated at x = 3$?

Accepted Answer

$$f(g(3)) = (2 	imes 3 + 1)^2$ = (6 + 1)^2$ = 49$$

Question 13

Which of the following is the equation of a line passing through the points $$(2, 5)$$ and $$(4, 9)$$?

Accepted Answer

$$y = 2x + 1$$

Question 14

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

Accepted Answer

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

Question 15

Which of the following is the correct domain for the function $$f(x) = \frac{1}{x - 4}$$?

Accepted Answer

All real numbers $$x$$ except $$x = 4$$

Question 16

A medical researcher uses the formula $$T = \frac{a + b}{2} to $$estimate the average temperature of two patients. If patient A has a temperature of $$98.6^\circF $$and patient B has $$101.2^\circF$$, what is the average temperature?

Accepted Answer

$$T = \frac{98.6 + 101.2}{2} = \frac{199.8}{2} = 99.9^\circF$$

Question 17

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

Accepted Answer

$$x = 32$$

Question 18

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

Accepted Answer

$$f(-3) = -3 + 2 = -1$$, $$f(2) = 2 	imes 2 - 1 = 3$$

College Algebra Syllabus and Core Competencies – Study Guide

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