Question 1

A pharmaceutical company is developing a new drug and needs to determine the concentration of the drug in the bloodstream over time. The concentration is modeled by the quadratic equation $$C(t) = -0.5t^2$ + 4t + 10$$, where $$C is $$the concentration in mg/L and $$t is $$time in hours. What is the maximum concentration of the drug, and at what time does it occur?

Accepted Answer

The maximum concentration is $$C = 14 mg/L at$$ $$t = 2$$ hours.

Question 2

A company wants to build a rectangular enclosure with a fixed perimeter of 60 meters. If the length is $$x$$ meters, express the area $$A as a $$function of $$x$$, and determine the value of $$x$$ that maximizes the area.

Accepted Answer

Area function: $$A(x) = x(30 - x)$$; maximum at $$x = 15$$ meters.

Question 3

Solve the quadratic equation $$2x^2 - 4x + 5 = 0$ using the quadratic formula. What are the solutions?

Accepted Answer

The solutions are $$x = 1 \pm \frac{i\sqrt{6}}{2}.$$

Question 4

A population of bacteria grows according to the exponential function $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the population after 5 hours?

Accepted Answer

The population is $$P(5) = 500e$$^{1}$$ \approx 1359.$$

Question 5

Which of the following is the correct interval notation for the set of all real numbers $$x$$ such that $$-3 \u003c x \leq 5$$?

Accepted Answer

$$(-3, 5]$$

Question 6

A piecewise function models the cost $$C(x) of $$shipping a package, where $$x is $$the weight in pounds:

$$C(x) = \begin{cases} 5 & 	ext{if } 0  5 \end{cases}$$

What is the cost to ship a package weighing 4 pounds?

Accepted Answer

$$C(4) = 8$$

Question 7

Given $$f(x) = 2x + 3$$ and $$g(x) = x^2$, what is the composite function f(g(x$$))$$?

Accepted Answer

$$f(g(x)) = 2x^2$ + 3$$

Question 8

A rational function is given by $$R(x) = \frac{x^2$ - 9}{x - 3}. $$What is the simplified form of $$R(x)$$ for $$x 
eq 3$$?

Accepted Answer

$$R(x) = x + 3$$

Question 9

Solve the absolute value equation $$|2x - 5| = 7$$ for $$x.$$

Accepted Answer

$$x = 6 or$$ $$x = -1$$

Question 10

A chemical reaction's rate is modeled by $$r(t) = 3t^2$ - 12t + 9. $$Find all values of $$t$$ where the rate is zero.

Accepted Answer

Question 11

A polynomial $$p(x) = x^3$ - 4x^2$ + x + 6 is $$divided by $$x - 2$$ using synthetic division. What is the remainder?

Accepted Answer

The remainder is $$10.$$

Question 12

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

Accepted Answer

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

Question 13

Which property of logarithms allows you to write $$\log_{b}(xy) = \log$$_{b}$$x + \log$$_{b}$$y$$?

Accepted Answer

Product property

Question 14

Solve the exponential equation $$2^{x} = 16$ for x$.

Accepted Answer

$$x = 4$$

Question 15

A quadratic function $$f(x) = x^2$ - 6x + 8 is $$transformed by shifting it 3 units up and 2 units right. What is the new function?

Accepted Answer

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

Question 16

A rational inequality $$\frac{x - 2}{x + 1} \u003e 0 is $$given. Which interval represents the solution set?

Accepted Answer

$$(2, \infty)$$

Question 17

A function $$f(x) is $$defined as $$f(x) = \sqrt{x - 4}. $$What is the domain of $$f(x) in $$interval notation?

Accepted Answer

$$[4, \infty)$$

Question 18

A medical test result is modeled by the function $$T(x) = \log_{10}(x)$$, where $$x is $$the concentration of a substance. If the concentration doubles, how does the test result change?

Accepted Answer

It increases by $$\log_{10}(2).$$

Question 19

A polynomial $$p(x) = x^4$ - 5x^2$ + 4 is $$factored. Which of the following is the correct complete factorization?

Accepted Answer

$$(x^2$$ - $$1)(x^2 - 4$$)$$

College Algebra: Course Structure and Key Topics Overview

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