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

$$\{ x \in \mathbb{R} \mid x \neq 3 \}$$

Q: A quadratic equation models the height $$h (in $$meters) of a ball thrown upward from a bridge as $$h(t) = -4.9t^2$ + 20t + 15. $$What is the maximum height reached by the ball?

$$h_{max} = h\left(\frac{-b}{2a}\right) = h\left(\frac{-20}{2 \times -4.9}\right) = h(2.04) \approx 35.41$$

Q: A biologist models the population of bacteria in a petri dish with $$P(t) = 200e$$^{0.3t}$$, where t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

$$P(5) = 200e$$^{1.5}$$ \approx 200 \times 4.4817 \approx 896$$

Question 1

A pharmaceutical company is testing a new drug and needs to calculate the average rate of change of the drug concentration in the bloodstream over time. If the concentration $$C(t) in mg/L at $$time $$t ($$hours) is given by $$C(t) = 5t^2$ + 2$$, what is the average rate of change of concentration from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$\frac{C(4) - C(1)}{4 - 1} = \frac{82 - 7}{3} = 25$$

Question 2

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

Accepted Answer

$$C = 30 + 0.20m$$

Question 3

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

Accepted Answer

$$\{ x \in \mathbb{R} \mid x 
eq 3 \}$$

Question 4

A quadratic equation models the height $$h (in $$meters) of a ball thrown upward from a bridge as $$h(t) = -4.9t^2$ + 20t + 15. $$What is the maximum height reached by the ball?

Accepted Answer

$$h_{max} = h\left(\frac{-b}{2a}ight) = h\left(\frac{-20}{2 	imes -4.9}ight) = h(2.04) \approx 35.41$$

Question 5

Simplify the expression $$\$$left(2x^3$$y^2$$\$$right)^2$.

Accepted Answer

$$4x^6y^4$

Question 6

A hospital uses the function $$f(x) = 100 - 2x to $$model the number of available beds after admitting $$x$$ patients. If $$f(x)$$ must be non-negative, what is the maximum number of patients that can be admitted?

Accepted Answer

$$x \leq 50$$

Question 7

Which of the following is the factored form of the polynomial $$x^2 - 5x + 6$?

Accepted Answer

$$(x - 2)(x - 3)$$

Question 8

A biologist models the population of bacteria in a petri dish with $$P(t) = 200e$$^{0.3t}$$, where t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

Accepted Answer

$$P(5) = 200e$$^{1.5}$$ \approx 200 	imes 4.4817 \approx 896$$

Question 9

Solve the quadratic equation $$x^2 + 4x + 5 = 0$ using the quadratic formula.

Accepted Answer

$$x = -2 \pm i$$

Question 10

A chemical reaction is modeled by the linear equation $$y = 3x + 7$$, where $$y is $$the concentration of a product and $$x is $$the time in minutes. What is the concentration after 10 minutes?

Accepted Answer

$$y = 3 	imes 10 + 7 = 37$$

Question 11

Which of the following is NOT a function?

Accepted Answer

A relation where $$x = 2$$ maps to both $$y = 3$$ and $$y = 5$$

Question 12

A student is solving the system of equations: $$\begin{cases} 2x + 3y = 12 \ x - y = 1 \end{cases}. $$What is the solution?

Accepted Answer

$$x = 3, y = 2$$

Question 13

Simplify the expression $$\frac{2x^2$y}{4xy^2$}.$$

Accepted Answer

$$\frac{x}{2y}$$

Question 14

A company’s profit is modeled by $$P(x) = -2x^2$ + 12x - 20$$, where $$x is $$the number of units sold. What is the minimum number of units that must be sold for the company to break even (profit $$P(x) = 0$$)?

Accepted Answer

$$x = 2 or$$ $$x = 5$$

Question 15

Which of the following correctly represents the interval notation for the solution to the inequality $$x \u003e -3$$?

Accepted Answer

$$(-3, \infty)$$

Question 16

A quadratic function $$f(x) = ax^2$ + bx + c$$ has its vertex at $$(2, -3). $$Which equation could represent this function?

Accepted Answer

$$f(x) = a(x - 2)^2$ - 3$$

Question 17

A medical researcher needs to solve $$\sqrt{x + 5} = 4$$ for $$x. $$What is the solution?

Accepted Answer

$$x = 11$$

Question 18

A student is asked to evaluate $$f(x) = 2x^2$ - 3x + 4 at$$ $$x = -2. $$What is the result?

Accepted Answer

$$f(-2) = 2(-2)^2$ - 3(-2) + 4 = 8 + 6 + 4 = 18$$

Question 19

Which of the following is the correct solution to the equation $$2x - 5 = 9$$?

Accepted Answer

$$x = 7$$

College Algebra Prep (MAT 114) Syllabus and Course Competencies Study Guide

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