Chapter 4

Fill in the table according to the given rule and find an expression for the function represented by the rule. An automobile purchased for \(\$ 20,000\) depreciates at a rate of \(10 \%\) per year. $$\begin{aligned} &\begin{array}{cc} \text { Years Since } & \underline{\phantom{xxx}} \\ \text { Purchase } & \text { Value } \end{array}\\\ &\begin{array}{l} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array} \end{aligned}$$

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=\log |x|$$

Applications In this set of exercises, you will use inverse functions to study real-world problems. When you buy products at a store, the Universal Product Code (UPC) is scanned and the price is output by a computer. The price is a function of the UPC. Why? Does this function have an inverse? Why or why not?

Fill in the table according to the given rule and find an expression for the function represented by the rule. A rain forest with a current area of 10,000 square kilometers loses 5% of its area every year. $$\begin{aligned} &\begin{array}{cc} \text { Years in the } & \text { Years in the } \\ \text { Future } & \text { Rainforest \(\left(\mathrm{km}^{2}\right)\) } \end{array}\\\ &\begin{array}{l} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array} \end{aligned}$$

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Find the hydrogen ion concentration of a solution with a \(\mathrm{pH}\) of 7.2.

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$g(x)=\ln \left(x^{2}\right)$$

Applications In this set of exercises, you will use inverse functions to study real-world problems. In economics, the demand function gives the price \(p\) as a function of the quantity \(q .\) One example of a demand function is \(p=100-0.1 q .\) However, mathematicians tend to think of the price as the input variable and the quantity as the output variable. How can you take this example of a demand function and express \(q\) as a function of p?

Suppose the population of a colony of bacteria doubles in 12 hours from an initial population of 1 million. Find the growth constant \(k\) if the population is modeled by the function \(P(t)=P_{0} e^{k t} .\) When will the population reach 4 million? 8 million?

The depreciation rate of a Mercury Sable is about \(30 \%\) per year. If the Sable was purchased for \(\$ 18,000,\) make a table of its values over the first 5 years after purchase. Find a function that gives its value \(t\) years after purchase, and sketch a graph of the function. (Source: Kelley Blue Book)

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Find the hydrogen ion concentration of a solution with a \(\mathrm{pH}\) of 3.4.

Applications In this set of exercises, you will use inverse functions to study real-world problems. After \(t\) seconds, the height of an object dropped from an initial height of 100 feet is given by \(h(t)=-16 t^{2}+100, t \geq 0\) (a) Why does \(h\) have an inverse? (b) Write \(t\) as a function of \(h\) and explain what it represents.

Solve each exponential equation. $$ 2^{x-1}=10 $$

The depreciation rate of a Toyota Camry is about 8% per year. If the Camry was purchased for $25,000, make a table of its values over the first 4 years after purchase. Find a function that gives its value years after purchase, and sketch a graph of the function. (Source: Kelley Blue Book)

Use the following information for Exercises \(74-76\) The decibel ( \(d B\) ) is a unit that is used to express the relative loudness of two sounds. One application of this is the relative value of the output power of an amplifier with respect to the input power. since power levels can vary greatly in magnitude, the relative value \(D\) of power level \(P_{1}\) with respect to power level \(P_{2}\) is given (in units of \(d B\) ) in terms of the logarithm of their ratio, as follows. $$D=10 \log \frac{P_{1}}{P_{2}}$$ The values \(P_{1}\) and \(P_{2}\) are expressed in the same units, such as watts \((W)\). If \(P_{1}=20 \mathrm{W}\) and \(P_{2}=0.3 \mathrm{W},\) find the relative value of \(P_{1}\) with respect to \(P_{2},\) in units of dB.

Applications In this set of exercises, you will use inverse functions to study real-world problems. A woman's dress size in the United States can be converted to a woman's dress size in France by using the function \(f(s)=s+30,\) where \(s\) takes on all even values from 2 to \(24,\) inclusive. (Source: www.onlineconversion \(. \operatorname{com})\) (a) What is the range of \(f ?\) (b) Find the inverse of \(f\) and interpret it.

In \(1965,\) Gordon Moore, then director of Intel research, conjectured that the number of transistors that fit on a computer chip doubles every few years. This has come to be known as Moore's Law. Analysis of data from Intel Corporation yields the following model of the number of transistors per chip over time: $$s(t)=2297.1 e^{0.3316 t}$$ where \(s(t)\) is the number of transistors per chip and \(t\) is the number of years since \(1971 .\) (Source: Intel Corporation) (a) According to this model, what was the number of transistors per chip in \(1971 ?\) (b) How long did it take for the number of transistors to double?

U.S. savings bonds, Series EE, pay interest at a rate of 3% compounded quarterly. How much would a bond purchased for $1000 be worth after 10 years? These bonds stop paying interest after 30 years. Why do you think this is so? (Hint: Think about how much this bond would be worth after 80 years.)

Use the following information for Exercises \(74-76\) The decibel ( \(d B\) ) is a unit that is used to express the relative loudness of two sounds. One application of this is the relative value of the output power of an amplifier with respect to the input power. since power levels can vary greatly in magnitude, the relative value \(D\) of power level \(P_{1}\) with respect to power level \(P_{2}\) is given (in units of \(d B\) ) in terms of the logarithm of their ratio, as follows. $$D=10 \log \frac{P_{1}}{P_{2}}$$ The values \(P_{1}\) and \(P_{2}\) are expressed in the same units, such as watts \((W)\). If an amplifier's output power is \(10 \mathrm{W}\) and the input power is \(0.5 \mathrm{W},\) what is the relative value of the output with respect to the input, in units of dB?

Applications In this set of exercises, you will use inverse functions to study real-world problems. When measuring temperature, \(100^{\circ}\) Celsius (C) is equivalent to \(212^{\circ}\) Fahrenhcit ( \(F\) ). Also, \(0^{\circ} \mathrm{C}\) is equivalent to \(32^{\circ} \mathrm{F}\) (a) Find a linear function that converts Celsius temperatures to Fahrenheit temperatures. (b) Find the inverse of the function you found in part (a). What does this inverse function accomplish?

The value of a 2003 Toyota Corolla is given by the function $$v(t)=14,000(0.93)^{t}$$ where \(t\) is the number of years since its purchase and \(v(t)\) is its value in dollars. (Source: Kelley Blue Book) (a) What was the Corolla's initial purchase price? (b) What percent of its value does the Toyota Corolla lose each year? (c) How long will it take for the value of the Toyota Corolla to reach \(\$ 12,000 \)

Use the following information for Exercises \(74-76\) The decibel ( \(d B\) ) is a unit that is used to express the relative loudness of two sounds. One application of this is the relative value of the output power of an amplifier with respect to the input power. since power levels can vary greatly in magnitude, the relative value \(D\) of power level \(P_{1}\) with respect to power level \(P_{2}\) is given (in units of \(d B\) ) in terms of the logarithm of their ratio, as follows. $$D=10 \log \frac{P_{1}}{P_{2}}$$ The values \(P_{1}\) and \(P_{2}\) are expressed in the same units, such as watts \((W)\). Use the properties of logarithms to show that the relative value of one power level with respect to another, expressed in units of \(\mathrm{dB},\) is actually a difference of two quantities.

The value of a 2006 S-type Jaguar is given by the function $$v(t)=43,173(0.8)^{t}$$ where \(t\) is the number of years since its purchase and \(v(t)\) is its value in dollars. (Source: Kelley Blue Book) (a) What was the Jaguar's initial purchase price? (b) What percentage of its value does the Jaguar S-type lose each year? (c) How many years will it take for the Jaguar S-type to reach a value of \(\$ 22,227 ?\)

The average hourly wage for construction workers was \(\$ 17.48\) in 2000 and has risen at a rate of \(2.7 \%\) annually. (Source: Bureau of Labor Statistics) (a) Find an expression for the average hourly wage as a function of time \(t .\) Measure \(t\) in years since 2000. (b) Using your answer to part (a), make a table of predicted values for the average hourly wage for the years 2000-2007. The actual average hourly wage for 2003 was \(\$ 18.95 .\) How does this value compare with the predicted value found in part (b)?

Consider the function \(f(x)=2^{x}\) (a) Sketch the graph of \(f\) (b) What are the domain and range of \(f ?\) (c) Graph the inverse function. (d) What are the domain and range of the inverse function?

Two students have an argument. One says that the inverse of the function \(f\) given by the expression \(f(x)=6\) is the function \(g\) given by the expression \(g(x)=\frac{1}{6} ;\) the other claims that \(f\) has no inverse. Who is correct and why?

The cumulative box office revenue from the movie Terminator 3 can be modeled by the logarithmic function $$R(x)=26.203 \ln x+90.798$$ where \(x\) is the number of weeks since the movie opened and \(R(x)\) is given in millions of dollars. How many weeks after the opening of the movie did the cumulative revenue reach \(\$ 140\) million? (Source: movies.yahoo.com)

When a drug is administered orally, the amount of the drug present in the bloodstream of the patient can be modeled by a function of the form $$C(t)=a t e^{-b t}$$ where \(C(t)\) is the concentration of the drug in milligrams per liter (mg/L), \(t\) is the number of hours since the drug was administered, and \(a\) and \(b\) are positive constants. For a 300 -milligram dose of the asthma drug aminophylline, this function is $$C(t)=4.5 t e^{-0.275 t}$$ (Source: Merck Manual of Diagnosis and Therapy) (a) How much of this drug is present in the bloodstream at time \(t=0 ?\) Why does this answer make sense in the context of the problem? (b) How much of this drug is present in the bloodstream after 1 hour? (c) Sketch a graph of this function, either by hand or using a graphing utility, with \(t\) ranging from 0 to \(20 .\) (d) What happens to the value of the function as \(t \rightarrow \infty ?\) Does this make sense in the context of the problem? Why? (e) Use a graphing utility to find the time when the concentration of this drug reaches its maximum. (f ) Use a graphing utility to determine when the concentration of this drug reaches 3 mg/L for the second time. (This will occur after the concentration peaks.)

Consider the function \(f(x)=x^{3}\) (a) Sketch the graph of \(f\) (b) What are the domain and range of \(f ?\) (c) Graph the inverse function. (d) What are the domain and range of the inverse function?

Do all linear functions have inverses? Explain.

Plutonium is a radioactive element that has a half-life of 24,360 years. The half-life of a radioactive substance is the time it takes for half of the substance to decay (which means the other half will still exist after that length of time). Find an exponential function of the form \(f(t)=A e^{k t}\) that gives the amount of plutonium left after \(t\) years if the initial amount of plutonium is 10 pounds. How long will it take for the plutonium to decay to 2 pounds?

The height (in feet) of the point on the Gateway Arch in Saint Louis that is directly above a given point along the base of the arch can be written as a function of the distance \(x\) (also in feet) of the latter point from the midpoint of the base: $$h(x)=-34.38\left(e^{-0.01 x}+e^{0.01 x}\right)+693.76$$ (Source: National Park Service) (Figure cant copy) (a) What is the maximum value of this function? (b) Evaluate \(h(100)\) (c) Graph the function \(h(x)\) using a graphing utility. Choose a suitable window size so that you can see the entire arch. For what value(s) of \(x\) is \(h(x)\) equal to 300 feet?

Graph \(f(x)=e^{\ln x}\) and \(g(x)=x\) on the same set of axes. (a) What are the domains of the two functions? (b) For what values of \(x\) do these two functions agree?

Solve each equation graphically and express the solution as an appropriate logarithm to four decimal places. If a solution does not exist, explain why. $$ 10^{t}=7 $$

If the graph of a function \(f\) is symmetric with respect to the \(y\) -axis, can \(f\) be one-to-one? Explain.

Refer to the following. The pH of a solution is defined as \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] .\) The concentration of hydrogen ions, \(\left[\mathrm{H}^{+}\right]\), is given in moles per liter, where one mole is equal to \(6.02 \times 10^{23}\) molecules. What is the concentration of hydrogen ions in a solution that has a pH of \(6.2 ?\)

In the definition of the exponential function, why is \(a=1\) excluded?

Graph \(f(x)=\ln e^{x}\) and \(g(x)=x\) on the same set of axes. (a) What are the domains of the two functions? (b) For what values of \(x\) do these two functions agree?

Solve each equation graphically and express the solution as an appropriate logarithm to four decimal places. If a solution does not exist, explain why. $$e^{t}=6$$

If a function \(f\) has an inverse and the graph of \(f\) lies in Quadrant IV, in which quadrant does the graph of \(f^{-1}\) lie?

Refer to the following. The pH of a solution is defined as \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] .\) The concentration of hydrogen ions, \(\left[\mathrm{H}^{+}\right]\), is given in moles per liter, where one mole is equal to \(6.02 \times 10^{23}\) molecules. What is the concentration of hydrogen ions in a solution that has a pH of \(1.5 ?\)

Consider the function \(f(x)=2+e^{-x}.\) (a) What number does \(f(x)\) approach as \(x \rightarrow+\infty ?\) (b) How could you use the graph of this function to confirm the answer to part (a)?

Let \(a > 1 .\) Can (-3,1) lie on the graph of \(\log _{a} x ?\) Why or why not?

Solve each equation graphically and express the solution as an appropriate logarithm to four decimal places. If a solution does not exist, explain why. $$4\left(10^{x}\right)=20$$

If a function \(f\) has an inverse and the graph of \(f\) lics in Quadrant III, in which quadrant does the graph of \(f^{-1}\) lie?

The 1960 earthquake in Chile registered 9.5 on the Richter scale. Find the energy \(E\) (in Ergs) released by using the following model, which relates the energy in Ergs to the magnitude \(R\) of an earthquake. (Source: National Earthquake Information Center, U.S. Geological Survey) $$\log E=11.4+(1.5) R$$

The graph of the function \(f(x)=C a^{x}\) passes through the points (0,12) and (2,3). (a) Use \(f(0)\) to find \(C.\) (b) Is this function increasing or decreasing? Explain. (c) Now that you know \(C\), use \(f(2)\) to find \(a\). Does your value of \(a\) confirm your answer to part (b)?

Solve each equation graphically and express the solution as an appropriate logarithm to four decimal places. If a solution does not exist, explain why. $$e^{t}=-3$$

Give an example of an odd function that is not one-to-one.

The decibel (dB) is a unit that is used to express the relative loudness of two sounds. One application of decibels is the relative value of the output power of an amplifier with respect to the input power. since power levels can vary greatly in magnitude, the relative value \(D\) of power level \(P_{1}\) with respect to power level \(P_{2}\) is given (in units of \(\mathrm{dB}\) ) in terms of the logarithm of their ratio as follows: $$D=10 \log \frac{P_{1}}{P_{2}}$$ where the values of \(P_{1}\) and \(P_{2}\) are expressed in the same units, such as watts \((\mathrm{W}) .\) If \(P_{2}=75 \mathrm{W},\) find the value of \(P_{1}\) at which \(D=0.7\)

Consider the two functions \(f(x)=2 x\) and \(g(x)=2^{x}.\) (a) Make a table of values for \(f(x)\) and \(g(x),\) with \(x\) ranging from -1 to 4 in steps of 0.5. (b) Find the interval(s) on which \(2 x<2^{x}.\) (c) Find the interval(s) on which \(2 x>2^{x}.\) (d) Using your table from part (a) as an aid, state what happens to the value of \(f(x)\) if \(x\) is increased by 1 unit. (e) Using your table from part (a) as an aid, state what happens to the value of \(g(x)\) if \(x\) is increased by 1 unit. (f) Using your answers from parts (c) and (d) as an aid, explain why the value of \(g(x)\) is increasing much faster than the value of \(f(x).\)