Question 1

Which of the following is the correct domain of the rational function $f(x) = \dfrac{2x+1}{x^2-9}$?

Accepted Answer

$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$

Question 2

A pharmaceutical company models the concentration $C$ of a drug in the bloodstream by $C(t) = \dfrac{50}{t+2}$, where $t$ is the time in hours after injection. What is the horizontal asymptote of this function, and what does it represent?

Accepted Answer

$y=0$, representing the drug concentration approaches zero as time increases.

Question 3

Given $f(x) = 3x - 7$, what is the inverse function $f^{-1}(x)$?

Accepted Answer

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

Question 4

Which of the following functions is one-to-one?

Accepted Answer

$f(x) = 2x + 5$

Question 5

A population of bacteria triples every 4 hours. Which equation models the population $P$ after $t$ hours if the initial population is 200?

Accepted Answer

$P(t) = 200 \cdot 3^{t/4}$

Question 6

Which of the following is the correct range of the exponential function $f(x) = 5e^{2x}$?

Accepted Answer

$(0, \infty)$

Question 7

Expand the logarithmic expression $\ln\left(\dfrac{4x^3}{y^2}\right)$ using properties of logarithms.

Accepted Answer

$\ln 4 + 3\ln x - 2\ln y$

Question 8

Condense the expression $2\log_{5}x - \dfrac{1}{2}\log_{5}y$ into a single logarithm.

Accepted Answer

$\log_{5}\left(\dfrac{x^2}{\sqrt{y}}\right)$

Question 9

Solve for $x$: $2^{x+1} = 16$

Accepted Answer

$x = 3$

Question 10

A radioactive substance decays according to $A(t) = A_0 e^{-0.03t}$, where $A_0$ is the initial amount and $t$ is in years. How long will it take for the substance to decay to half its original amount?

Accepted Answer

$t = \dfrac{\ln 2}{0.03}$

Question 11

If $f(x) = \dfrac{1}{x-2}$, which of the following statements is true about its asymptotes?

Accepted Answer

There is a vertical asymptote at $x=2$ and a horizontal asymptote at $y=0$.

Question 12

Given $f(x) = 2x + 1$ and $g(x) = \dfrac{x-1}{2}$, what is $f(g(x))$?

Accepted Answer

$f(g(x)) = x$

Question 13

Solve for $x$: $\log_{3}(x-2) = 2$

Accepted Answer

$x = 11$

Question 14

Which of the following is equivalent to $e^{\ln 7}$?

Accepted Answer

$7$

Question 15

A certain medicine is eliminated from the body at a rate modeled by $M(t) = 100e^{-0.2t}$, where $M$ is the amount in mg and $t$ is in hours. How much medicine remains after 5 hours?

Accepted Answer

$100e^{-1}$ mg$

Question 16

Which of the following is the solution to the equation $\log_{2}(x) + \log_{2}(x-3) = 3$?

Accepted Answer

$x = 4$

Question 17

If $f(x) = x^2$ and $g(x) = \sqrt{x}$, which of the following statements is true?

Accepted Answer

$f$ and $g$ are not inverses because $f(g(x)) = x$, but $g(f(x))$ is not always $x$.

Question 18

Use the change-of-base formula to evaluate $\log_{7} 49$.

Accepted Answer

$\dfrac{\ln 49}{\ln 7} = 2$

Question 19

Solve for $x$: $e^{2x} = 20$

Accepted Answer

$x = \dfrac{1}{2} \ln 20$

College Algebra: Rational, Inverse, Exponential, and Logarithmic Functions Study Guide

Study Guide - Practice Questions

Study Guide - Flashcards