Chapter Test

1. Find the inverse of the relation

${(- 2, 5), (4, 3), (0, - 1), (- 6, - 3} .$ ${(- 2, 5), (4, 3), (0, - 1), (- 6, - 3} .$

Determine whether the function is one-to-one. Answer yes or no.

In Exercises 4–7, given the function:

4. $f (x) = x^{3} + 1$ $f (x) = x^{3} + 1$
5. $f (x) = 1 - x$ $f (x) = 1 - x$
6. $f (x) = \frac{x}{2 - x}$ $f (x) = \frac{x}{2 - x}$
7. $f (x) = x^{2} + x - 3$ $f (x) = x^{2} + x - 3$
8. Use composition of functions to show that $f^{^{- 1}}$ $f^{^{- 1}}$ is as given:

$f (x) = - 4 x + 3, f^{- 1} (x) = \frac{3 - x}{4} .$ $f (x) = - 4 x + 3, f^{- 1} (x) = \frac{3 - x}{4} .$
9. Find the inverse of the one-to-one function

$f (x) = \frac{1}{x - 4} .$ $f (x) = \frac{1}{x - 4} .$

Give the domain and the range of f and of $f^{- 1}$ $f^{- 1}$ and then graph both f and $f^{- 1}$ $f^{- 1}$ on the same set of axes.

Graph the function.

Find each of the following. Do not use a calculator.

Find each of the following using a calculator. Round to four decimal places.

20. $\ln 16$ $\ln 16$
21. $\log 0.293$ $\log 0.293$
22. Find $\log_{6} 10$ $\log_{6} 10$ using the change-of-base formula.
23. Express as a single logarithm:

$2 \log_{a} x - \log_{a} y + \frac{1}{2} \log_{a} z .$ $2 \log_{a} x - \log_{a} y + \frac{1}{2} \log_{a} z .$
24. Express $\ln \sqrt[5]{x^{2} y}$ $\ln \sqrt[5]{x^{2} y}$ in terms of sums and differences of logarithms.
25. Given that $\log_{a} 2 = 0.328$ $\log_{a} 2 = 0.328$ and $\log_{a} 8 = 0.984$ $\log_{a} 8 = 0.984$ , find $\log_{a} 4$ $\log_{a} 4$ .
26. Simplify: $\ln e^{- 4 t}$ $\ln e^{- 4 t}$ .

Solve.

27. $\log_{25} 5 = x$ $\log_{25} 5 = x$
28. $\log_{3} x + \log_{3} (x + 8) = 2$ $\log_{3} x + \log_{3} (x + 8) = 2$
29. $3^{4 - x} = 27^{x}$ $3^{4 - x} = 27^{x}$
30. $e^{x} = 65$ $e^{x} = 65$
31. Earthquake Magnitude. The earthquake in Bam, in southeast Iran, on December 26, 2003, had an intensity of $10^{6.6} \cdot I_{0}$ $10^{6.6} \cdot I_{0}$ (Source: U.S. Geological Survey). What was its magnitude on the Richter scale?
32. Growth Rate. A country’s population doubled in 45 years. What was the exponential growth rate?
33. Compound Interest. Suppose $1000 is invested at interest rate k, compounded continuously, and grows to $1144.54 in 3 years.
1. Find the interest rate.
2. Find the exponential growth function.
3. Find the balance after 8 years.
4. Find the doubling time.
34. The graph of $f (x) = 2^{x - 1} + 1$ $f (x) = 2^{x - 1} + 1$ is which of the following?

Synthesis

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