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

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