Q: A pharmaceutical company models the concentration of a drug in the bloodstream by $C(t) = 50e^{-0.4t}$, where $t$ is in hours. How long will it take for the concentration to drop to $10$ mg/L?

$t = \dfrac{\ln(5)}{0.4} \approx 4.02$ hours

Q: A business’s cost function is $C(x) = 5x^2 + 200x + 1000$. What is the average cost per item when $x = 20$?

$\dfrac{C(20)}{20} = \dfrac{5000 + 4000 + 1000}{20} = \dfrac{10000}{20} = 500$

Q: Which of the following statements is true about the function $f(x) = x^2$?

Its graph is a parabola opening upward.

Q: If $f(x) = 2x + 3$ and $g(x) = x^2$, what is $(f \circ g)(2)$?

$(f \circ g)(2) = f(g(2)) = f(4) = 2 \times 4 + 3 = 11$

Q: A scientist uses the Richter scale, which is logarithmic, to measure earthquake intensity. If one earthquake measures $M = 6$ and another $M = 8$, how many times more intense is the second earthquake?

$10^{2} = 100$ times more intense

Q: A savings account earns $4\%$ interest compounded continuously. If the initial deposit is $\$2000$, what is the balance after $10$ years? (Use $A = Pe^{rt}$)

$A = 2000e^{0.04 \times 10} = 2000e^{0.4} \approx 2982$

Question 1

A company’s revenue $R$ (in dollars) from selling $x$ units of a product is given by the linear function $R(x) = 25x + 400$. What is the slope of the revenue function, and what does it represent in this context?

Accepted Answer

The slope is $25$, representing the revenue generated per unit sold.

Question 2

Given the quadratic function $f(x) = 2x^2 - 8x + 6$, what are the coordinates of the vertex?

Accepted Answer

The vertex is at $(2, -2)$.

Question 3

A population of bacteria grows exponentially according to $P(t) = 500e^{0.3t}$, where $t$ is in hours. What is the population after $5$ hours?

Accepted Answer

$P(5) = 500e^{1.5} \approx 2240$

Question 4

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

Accepted Answer

$x 
eq 4$

Question 5

A car’s value depreciates exponentially according to $V(t) = 20000e^{-0.15t}$, where $t$ is the number of years after purchase. What is the value of the car after $8$ years?

Accepted Answer

$V(8) = 20000e^{-1.2} \approx 6020$

Question 6

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

Accepted Answer

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

Question 7

Given the piecewise function $f(x) = \begin{cases} x^2 & x < 0 \ 2x + 1 & x \geq 0 \end{cases}$, what is $f(-3) + f(2)$?

Accepted Answer

$f(-3) + f(2) = 9 + 5 = 14$

Question 8

A pharmaceutical company models the concentration of a drug in the bloodstream by $C(t) = 50e^{-0.4t}$, where $t$ is in hours. How long will it take for the concentration to drop to $10$ mg/L?

Accepted Answer

$t = \dfrac{\ln(5)}{0.4} \approx 4.02$ hours

Question 9

Which of the following is the correct solution to the equation $2^{x} = 10$?

Accepted Answer

$x = \log_{2}{10}$

Question 10

A business’s cost function is $C(x) = 5x^2 + 200x + 1000$. What is the average cost per item when $x = 20$?

Accepted Answer

$\dfrac{C(20)}{20} = \dfrac{5000 + 4000 + 1000}{20} = \dfrac{10000}{20} = 500$

Question 11

Which of the following statements is true about the function $f(x) = x^2$?

Accepted Answer

Its graph is a parabola opening upward.

Question 12

If $f(x) = 2x + 3$ and $g(x) = x^2$, what is $(f \circ g)(2)$?

Accepted Answer

$(f \circ g)(2) = f(g(2)) = f(4) = 2 	imes 4 + 3 = 11$

Question 13

A chemical reaction’s rate is modeled by $R(t) = 100e^{-0.2t}$. What is the limiting value of $R(t)$ as $t 	o \infty$?

Accepted Answer

$0$

Question 14

Which of the following is the correct solution to the quadratic equation $x^2 - 6x + 5 = 0$?

Accepted Answer

$x = 1$ and $x = 5$

Question 15

A scientist uses the Richter scale, which is logarithmic, to measure earthquake intensity. If one earthquake measures $M = 6$ and another $M = 8$, how many times more intense is the second earthquake?

Accepted Answer

$10^{2} = 100$ times more intense

Question 16

Which of the following is the correct equation for the line passing through $(2, 5)$ with a slope of $-3$?

Accepted Answer

$y = -3x + 11$

Question 17

A savings account earns $4\%$ interest compounded continuously. If the initial deposit is $\$2000$, what is the balance after $10$ years? (Use $A = Pe^{rt}$)

Accepted Answer

$A = 2000e^{0.04 	imes 10} = 2000e^{0.4} \approx 2982$

Question 18

Which of the following is true about the function $f(x) = \ln(x)$?

Accepted Answer

Its domain is $x > 0$.

College Algebra: Core Concepts, Functions, and Applications – Study Guide

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