Question 1

Which of the following functions is one-to-one on its entire domain?

Accepted Answer

$$y = x^3$

Question 2

Given $$f(x) = e^x$$, what is the domain and range of its inverse function?

Accepted Answer

Domain: $$(0, \infty)$$, Range: $$(-\infty, \infty)$$

Question 3

Which property of logarithms allows you to write $$\ln(ab) as$$ $$\ln a + \ln b$$?

Accepted Answer

Product Rule

Question 4

If $$f(x) = x^3$ + 2$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \sqrt[3]{x - 2}$$

Question 5

Which of the following statements is TRUE about the graph of a function and its inverse?

Accepted Answer

The graph of $$f^{-1}(x$$)$$ is a reflection of f(x$$)$$ about the line y = x$.

Question 6

Expand $$\ln\left(\frac{x^3$}{y}\right)$$ using properties of logarithms.

Accepted Answer

$$3\ln x - \ln y$$

Question 7

Which of the following is the correct domain for the function $$f(x) = \sqrt{x}$$?

Accepted Answer

$$[0, \infty)$$

Question 8

Suppose $$f(x) is $$one-to-one with domain $$D$$ and range $$R. $$What is the domain and range of $$f^{-1}(x$$)$$?

Accepted Answer

Domain: $$R$$, Range: $$D$$

Question 9

Condense $$2\ln x + \frac{1}{2}\ln y$$ into a single logarithm.

Accepted Answer

$$\ln(x^2$y^{1/2}$$)$

Question 10

Which of the following is NOT a property of the natural logarithm?

Accepted Answer

$$\ln(a + b) = \ln a + \ln b$$

Question 11

If $$f(x) = \sin x$$, why does $$f$$ not have an inverse on $$(-\infty, \infty)$$?

Accepted Answer

Because $$\sin x is $$not one-to-one on $$(-\infty, \infty).$$

Question 12

Which of the following best describes the range of $$f(x) = \ln x$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 13

If $$f(x) = 2x + 5$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 5}{2}$$

Question 14

Which of the following is the correct inverse of $$f(x) = \sqrt{x}$$ for $$x \geq 0$$?

Accepted Answer

$$f^{-1}(x$$) = $$x^2$$

Question 15

Which of the following is the correct application of the Power Rule for logarithms?

Accepted Answer

$$\ln(x^n) = n\ln x$$

Question 16

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-kt}$$. If N_0$ = 100$$ and $$k = 0.1$$, how long until only 25 units remain?

Accepted Answer

$$t = \frac{\ln 4}{0.1}$$

Question 17

Which of the following is the correct condensed form of $$\ln x + \ln y - \ln z$$?

Accepted Answer

$$\ln\left(\frac{xy}{z}\right)$$

Question 18

If $$f(x) = \ln(x-2)$$, what is the domain of $$f$$?

Accepted Answer

$$(2, \infty)$$

Question 19

Which of the following is the correct inverse of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{1}{2}\ln x$$

Inverse Functions and Logarithms: Study Guide

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