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 $-3 < x \leq 5$?

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:

$C(x) = \begin{cases} 5 & 	ext{if } 0 < x \leq 2 \ 8 & 	ext{if } 2 < x \leq 5 \ 12 & 	ext{if } x > 5 \end{cases}$

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.

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 $\frac{x - 2}{x + 1} > 0$ 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

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

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