Mid-Chapter Mixed Review

Determine whether the statement is true or false.

The domain of all logarithmic functions is $[1, \infty)$ $[1, \infty)$ . [5.3]
The range of a one-to-one function f is the domain of its inverse $f^{- 1}$ $f^{- 1}$ . [5.1]
The y-intercept of $f (x) = e^{- x}$ $f (x) = e^{- x}$ is (0, −1). [5.2]

For each function, determine whether it is one-to-one, and if the function is one-to-one, find a formula for its inverse. [5.1]

$f (x) = - \frac{2}{x}$ $f (x) = - \frac{2}{x}$
$f (x) = 3 + x^{2}$ $f (x) = 3 + x^{2}$
$f (x) = \frac{5}{x - 2}$ $f (x) = \frac{5}{x - 2}$
Given the function $f (x) = \sqrt{x - 5}$ $f (x) = \sqrt{x - 5}$ , use composition of functions to show that $f^{- 1} (x) = x^{2} + 5$ $f^{- 1} (x) = x^{2} + 5$ . [5.1]
Given the one-to-one function $f (x) = x^{3} + 2$ $f (x) = x^{3} + 2$ , find the inverse, give the domain and the range of f and $f^{- 1}$ $f^{- 1}$ , and graph both f and $f^{- 1}$ $f^{- 1}$ on the same set of axes. [5.1]

Match the function with one of the graphs (a)–(h) that follow. [5.2], [5.3]

$y = \log_{2} x$ $y = \log_{2} x$
$f (x) = 2^{x} + 2$ $f (x) = 2^{x} + 2$
$f (x) = e^{x - 1}$ $f (x) = e^{x - 1}$
$f (x) = \ln x - 2$ $f (x) = \ln x - 2$
$f (x) = \ln (x - 2)$ $f (x) = \ln (x - 2)$
$y = 2^{- x}$ $y = 2^{- x}$
$f (x) = | \log x |$ $f (x) = | \log x |$
$f (x) = e^{x} + 1$ $f (x) = e^{x} + 1$
Suppose that $3200 is invested at $4 \frac{1}{2} %$ $4 \frac{1}{2} %$ interest, compounded quarterly. Find the amount of money in the account in 6 years. [5.2]

Find each of the following without a calculator. [5.3]

$\log_{4} 1$ $\log_{4} 1$
$\ln e^{- 4 / 5}$ $\ln e^{- 4 / 5}$
$\log 0.01$ $\log 0.01$
$\ln e^{2}$ $\ln e^{2}$
$\ln 1$ $\ln 1$
$\log_{2} \frac{1}{16}$ $\log_{2} \frac{1}{16}$
$\log 1$ $\log 1$
$\log_{3} 27$ $\log_{3} 27$
$\log \sqrt[4]{10}$ $\log \sqrt[4]{10}$
$\ln e$ $\ln e$
Convert $e^{- 6} = 0.0025$ $e^{- 6} = 0.0025$ to a logarithmic equation. [5.3]
Convert $\log T = r$ $\log T = r$ to an exponential equation. [5.3]

Find the logarithm using the change-of-base formula. [5.3]

Collaborative Discussion and Writing

Explain why an even function f does not have an inverse $f^{- 1}$ $f^{- 1}$ that is a function. [5.1]
Suppose that $10,000 is invested for 8 years at 6.4% interest, compounded annually. In what year will the most interest be earned? Why? [5.2]
Describe the differences between the graph of $f (x) = x^{3}$ $f (x) = x^{3}$ and the graph of $g (x) = 3^{x}$ $g (x) = 3^{x}$ . [5.2]
If $\log b < 0$ $\log b < 0$ , what can you say about b? [5.3]

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