In 2 4, for the given functions f and g find: (a) 1f g2

Evaluate each expression using the graphs of y = f1x2 and y = g1x2 shown in the figure. x y 10 8 6 4 2 2468 10 6 4 2 4 6 2 y = f(x) (8, 2) (5, 4) (8, 2) (2, 1) (0, 3) (2, 0) (4, 3) (7, 6) (10, 3) (10, 2) (7, 0) (4, 6) (2, 4) y = g(x) (0, 6) (5, 4) (a) 1g  f2 1 -82 (b) 1f  g2 1 -82 (c) 1g  g2 172 (d) 1g  f2 1 -52

In Problems 2 4, for the given functions f and g find: (a) 1f  g2 122 (b) 1g  f2 1 -22 (c) 1f  f2 142 (d) 1g  g2 1 -12 2. f1x2 = 3x - 5; g1x2 = 1 - 2x2 3. f1x2 = 2x + 2: g1x2 = 2x2 + 1 4. f1x2 = ex ; g1x2 = 3x - 2

In Problems 2 4, for the given functions f and g find: (a) 1f  g2 122 (b) 1g  f2 1 -22 (c) 1f  f2 142 (d) 1g  g2 1 -12 2. f1x2 = 3x - 5; g1x2 = 1 - 2x2 3. f1x2 = 2x + 2: g1x2 = 2x2 + 1 4. f1x2 = ex ; g1x2 = 3x - 3

In Problems 2 4, for the given functions f and g find: (a) 1f  g2 122 (b) 1g  f2 1 -22 (c) 1f  f2 142 (d) 1g  g2 1 -12 2. f1x2 = 3x - 5; g1x2 = 1 - 2x2 3. f1x2 = 2x + 2: g1x2 = 2x2 + 1 4. f1x2 = ex ; g1x2 = 3x - 4

In Problems 57, find f  g, g  f, f  f, and g  g for each pair of functions. State the domain of each composite function.

In Problems 57, find f  g, g  f, f  f, and g  g for each pair of functions. State the domain of each composite function.

In Problems 57, find f  g, g  f, f  f, and g  g for each pair of functions. State the domain of each composite function.

In Problem 8, (a) verify that the function is one-to-one, and (b) find the inverse of the given function. 8. {11, 22, 13, 52, 15, 82, 16, 102}

In Problem 9, state why the graph of the function is one-toone. Then draw the graph of the inverse function f -1 . For convenience (and as a hint), the graph of y = x is also given.

In Problems 1013, the function f is one-to-one. Find the inverse of each function and check your answer. State the domain and the range of f and f -1.

In Problems 1013, the function f is one-to-one. Find the inverse of each function and check your answer. State the domain and the range of f and f -1.

In Problems 1013, the function f is one-to-one. Find the inverse of each function and check your answer. State the domain and the range of f and f -1.

In Problems 1013, the function f is one-to-one. Find the inverse of each function and check your answer. State the domain and the range of f and f -1.

Evaluate: (a) f142 (b) g192 (c) f(-22 (d) ga 1 27 b

Convert 52 = z to an equivalent statement involving a logarithm

Convert log5 u = 13 to an equivalent statement involving an exponent.

In Problems 17 and 18, find the domain of each logarithmic function

In Problems 17 and 18, find the domain of each logarithmic function

In Problems 1921, evaluate each expression. Do not use a calculator

In Problems 1921, evaluate each expression. Do not use a calculator

In Problems 1921, evaluate each expression. Do not use a calculator

In Problems 2225, write each expression as the sum and/or difference of logarithms. Express powers as factors.

In Problems 2225, write each expression as the sum and/or difference of logarithms. Express powers as factors.

In Problems 2225, write each expression as the sum and/or difference of logarithms. Express powers as factors.

In Problems 2225, write each expression as the sum and/or difference of logarithms. Express powers as factors.

In Problems 2628, write each expression as a single logarithm.

In Problems 2628, write each expression as a single logarithm.

In Problems 2628, write each expression as a single logarithm.

Use the Change-of-Base Formula and a calculator to evaluate log4 19. Round your answer to three decimal places.

Graph y = log3 x using a graphing utility and the Change-of-Base Formula.

In Problems 3134, use the given function f to: (a) Find the domain of f. (b) Graph f. (c) From the graph, determine the range and any asymptotes of f. (d) Find f -1, the inverse of f. (e) Find the domain and the range of f -1 . (f) Graph f -1.

In Problems 3134, use the given function f to: (a) Find the domain of f. (b) Graph f. (c) From the graph, determine the range and any asymptotes of f. (d) Find f -1, the inverse of f. (e) Find the domain and the range of f -1 . (f) Graph f -1.

In Problems 3134, use the given function f to: (a) Find the domain of f. (b) Graph f. (c) From the graph, determine the range and any asymptotes of f. (d) Find f -1, the inverse of f. (e) Find the domain and the range of f -1 . (f) Graph f -1.

In Problems 3134, use the given function f to: (a) Find the domain of f. (b) Graph f. (c) From the graph, determine the range and any asymptotes of f. (d) Find f -1, the inverse of f. (e) Find the domain and the range of f -1 . (f) Graph f -1.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

In Problems 3545, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility.

Suppose that f1x2 = log2 1x - 22 + 1. (a) Graph f. (b) What is f 162? What point is on the graph of f ? (c) Solve f1x2 = 4. What point is on the graph of f ? (d) Based on the graph drawn in part (a), solve f1x2 7 0. (e) Find f -1 1x2. Graph f -1 on the same Cartesian plane as f.

An amplifiers power output P (in watts) is related to its decibel voltage gain d by the formula P = 25e0.1d (a) Find the power output for a decibel voltage gain of 4 decibels. (b) For a power output of 50 watts, what is the decibel voltage gain?

A telescope is limited in its usefulness by the brightness of the star that it is aimed at and by the diameter of its lens. One measure of a stars brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L of a Chapter Review 349 350 CHAPTER 5 Exponential and Logarithmic Functions telescope, that is, the magnitude of the dimmest star that it can be used to view, is given by L = 9 + 5.1 log d where d is the diameter (in inches) of the lens. (a) What is the limiting magnitude of a 3.5-inch telescope? (b) What diameter is required to view a star of magnitude 14?

The number of years n for a piece of machinery to depreciate to a known salvage value can be found using the formula n = log s - log i log(1 - d) where s is the salvage value of the machinery, i is its initial value, and d is the annual rate of depreciation. (a) How many years will it take for a piece of machinery to decline in value from $90,000 to $10,000 if the annual rate of depreciation is 0.20 (20%)? (b) How many years will it take for a piece of machinery to lose half of its value if the annual rate of depreciation is 15%?

A childs grandparents purchase a $10,000 bond fund that matures in 18 years to be used for her college education. The bond fund pays 4% interest compounded semiannually. How much will the bond fund be worth at maturity? What is the effective rate of interest? How long will it take the bond to double in value under these terms?

A childs grandparents wish to purchase a bond that matures in 18 years to be used for her college education. The bond pays 4% interest compounded semiannually. How much should they pay so that the bond will be worth $85,000 at maturity?

The bones of a prehistoric man found in the desert of New Mexico contain approximately 5% of the original amount of carbon 14. If the half-life of carbon 14 is 5700 years, approximately how long ago did the man die?

A skillet is removed from an oven whose temperature is 450F and placed in a room whose temperature is 70F. After 5 minutes, the temperature of the skillet is 400F. How long will it be until its temperature is 150F?

The annual growth rate of the worlds population in 2010 was k = 1.15% = 0.0115. The population of the world in 2010 was 6,852,472,823. Letting t = 0 represent 2010, use the uninhibited growth model to predict the worlds population in the year 2015

In fiscal year 2010, the federal deficit was $1.3 trillion. At that time, 10-year treasury notes were paying 3.35% interest per annum. If the federal government financed this deficit through 10-year notes, how much would it have to pay back in 2020? How much was the interest?

The logistic growth model P1t2 = 0.8 1 + 1.67e-0.16t

The following data were collected by placing a temperature probe in a portable heater, removing the probe, and then recording temperature over time. Time (sec.) Temperature (F) 0 1 2 9 10 11 12 165.07 164.77 159.35 158.61 157.89 13 14 156.11 156.83 155.08 154.40 15 3 4 5 6 7 8 153.72 3 163.99 163.22 162.82 161.96 161.20 160.45 According to Newtons Law of Cooling, these data should follow an exponential model. (a) Using a graphing utility, draw a scatter diagram for the data. (b) Using a graphing utility, build an exponential model from the data. (c) Graph the exponential function found in part (b) on the scatter diagram. (d) Predict how long it will take for the probe to reach a temperature of 110F.

The data on the next page represent the wind speed (mph) and wind chill factor at an air temperature of 15F. (a) Using a graphing utility, draw a scatter diagram with wind speed as the independent variable. (b) Using a graphing utility, build a logarithmic model from the data. (c) Using a graphing utility, draw the logarithmic function found in part (b) on the scatter diagram. (d) Use the function found in part (b) to predict the wind chill factor if the air temperature is 15F and the wind speed is 23 mph.

Jack and Diane live in a small town of 50 people. Unfortunately, both Jack and Diane have a cold. Those who come in contact with someone who has this cold will themselves catch the cold. The following data represent the number of people in the small town who have caught the cold after t days. (a) Using a graphing utility, draw a scatter diagram of the data. Comment on the type of relation that appears to exist between the days and number of people with a cold. (b) Using a graphing utility, build a logistic model from the data.