Chapter 5

Some workers use technology such as e-mail, computers, and multiple phone lines to work at home, rather than in the office. However, because of the need for teamwork and collaboration in the workplace, fewer employees are telecommuting than expected. The table lists telecommuters \(T\) in millions during year \(x\). $$\begin{array}{cccccc} x & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline T & 9.2 & 9.6 & 10.0 & 10.4 & 10.6 \end{array}$$ $$\begin{array}{cccccc} \hline x & 2002 & 2003 & 2004 & 2005 & 2006 \\ T & 11.0 & 11.1 & 11.2 & 11.3 & 11.4\end{array}$$ Find a function \(f\) that models the data, where \(x=1\) corresponds to \(1997, x=2\) to \(1998,\) and so on.

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log (x+3)$$

Simplify the expression without a calculator $$ \left(5^{101}\right)^{1 / 101} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x^{2}-1, \quad g(x)=x^{2}+1 $$

The table shows the air pressure \(y\) in inches of mercury \(x\) miles from the eye of a hurricane. $$\begin{array}{ccccccc} \hline x & 2 & 4 & 8 & 15 & 30 & 100 \\ \hline y & 27.3 & 27.7 & 28.04 & 28.3 & 28.7 & 29.3\end{array}$$ (a) Make a scatterplot of the data. (b) Find a function \(f\) that models the data. (c) Estimate the air pressure at 50 miles.

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\ln (2 x-4)$$

Simplify the expression without a calculator $$ \left(8^{27}\right)^{1 / 27} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=4 x^{3}-8 x^{2}, \quad g(x)=4 x^{2} $$

The table lists the atmospheric density \(y\) in kilograms per cubic meter \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\) at an altitude of \(x\) meters. $$\begin{array}{ccccc} x(m) & 0 & 5000 & 10,000 & 15,000 \\ y\left(k g / m^{3}\right) & 1.2250 & 0.7364 & 0.4140 & 0.1948 \end{array}$$ $$\begin{array}{rllll} \boldsymbol{x}(\mathrm{m}) & 20,000 & 25,000 & 30,000 \\\ y\left(\mathrm{kg} / \mathrm{m}^{3}\right) & 0.0889 & 0.0401 & 0.0184 \end{array}$$ (a) Find a function \(f\) that models the data. (b) Prodict the density at 7000 meters. (The actual value is \(.0.59 \mathrm{kg} / \mathrm{m}^{3} .\))

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)-\log _{2}\left(x^{2}-1\right)$$

(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & 2 & 0.8 & -0.4 & -1.6 & -2.8 \end{array} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x-\sqrt{x-1}, \quad g(x)=x+\sqrt{x-1} $$

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log _{4}\left(4-x^{2}\right)$$

(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ y & 2 & 8 & 32 & 128 & 512 \end{array} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=3+\sqrt{2 x+9}, \quad g(x)=3-\sqrt{2 x+9} $$

The table at the top of the next column shows the density \(y\) of a species of insect measured in thousands per acre after \(x\) days. $$\begin{array}{cccccccc}x & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\ \hline y & 0.38 & 1.24 & 2.86 & 4.22 & 4.78 & 4.94 & 4.98 \end{array}$$ (a) Find a function \(f\) that models the data. (b) Use \(f\) to estimate the insect density after a long time.

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log _{3}\left(4^{x}\right)$$

(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & -3 & -2 & -1 & 0 & 1 \\ \hline y & 64 & 32 & 16 & 8 & 4 \end{array} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\sqrt{x}-1, \quad g(x)=\sqrt{x}+1 $$

As age increases, so does the likelihood of coronary heart disease (CHD). The percentage \(P\) of people \(x\) years old with signs of CHD is shown in the table. $$\begin{array}{rrrrrrrr}x & 15 & 25 & 35 & 45 & 55 & 65 & 75 \\ \hline P(\%) & 2 & 7 & 19 & 43 & 68 & 82 & 87\end{array}$$ (a) Evaluate \(P(25)\) and interpret the answer. (b) Find a function that models the data. (c) Graph \(P\) and the data. (d) At what age does a person have a \(50 \%\) chance of having signs of CHD?

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log _{5}\left(5^{x}-25\right)$$

(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & -2 & -1 & 0 & 1 & 2 \\ y & 3 & 5.5 & 8 & 10.5 & 13 \end{array} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\sqrt{1-x}, \quad g(x)=x^{3} $$

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\ln (\sqrt{3-x}-1)$$

(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & -4 & -2 & 0 & 2 & 4 \\ \hline y & 0.3125 & 1.25 & 5 & 20 & 80 \end{array} $$

The table is a complete representation of \(f\). Use the table to determine if \(f\) is one-to-one and has an inverse. \(\begin{array}{rrrrr}x & 1 & 2 & 3 & 4 \\ f(x) & 4 & 3 & 3 & 5\end{array}\)

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\frac{1}{x+1}, \quad g(x)=\frac{3}{x+1} $$

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log (4-\sqrt{2-x})$$

(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & -15 & -5 & 5 & 15 & 25 \\ \hline y & 22 & 24 & 26 & 28 & 30 \end{array} $$

The table is a complete representation of \(f\). Use the table to determine if \(f\) is one-to-one and has an inverse. \(\begin{array}{rrrrr}x & -2 & 0 & 2 & 4 \\ f(x) & 4 & 2 & 0 & -2\end{array}\)

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x^{1 / 2}, \quad g(x)=3 $$

Heavier birds tend to have larger wings than smaller birds. For one species of bird, the table lists the area \(A\) of the bird's wing in square inches if the bird weighs \(w\) pounds. $$\begin{array}{rccccc}w(\mathrm{b}) & 2 & 6 & 10 & 14 & 18 \\\\\hline A(w)\left(\mathrm{in}^{2}\right) & 160 & 330 & 465 & 580 & 685\end{array}$$ (a) Find a function that models the data. (b) Graph \(A\) and the data. (c) What weight corresponds to a wing area of 500 square inches?

Simplify the expression. $$\log _{8} 8^{-5.7}$$

Job Offer A company offers a college graduate \(\$ 40,000\) per year with a guaranteed \(8 \%\) raise each year. Is this an example of linear or exponential growth? Find a function \(f\) that computes the salary during the \(n\) th year.

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\frac{1}{2 x-4}, \quad g(x)=\frac{x}{2 x-4} $$

Heavier birds tend to have a longer wing span than smaller birds. For one species of bird, the table lists the length \(L\) of the bird's wing span in feet if the bird weighs w pounds. $$\begin{array}{rrrrr}w(\mathrm{b}) & 0.22 & 0.88 & 1.76 & 2.42 \\\L(w)(\mathrm{ft}) & 1.38 & 2.19 & 2.76 & 3.07\end{array}$$ (a) Find a function that models the data. (b) Graph \(L\) and the data. (c) What weight corresponds to a wing span of 2 feet?

Simplify the expression. $$ \log _{4} 4^{-1.23} $$

Job Offer A new employee is offered \(\$ 35,000\) per year with a guaranteed \(\$ 5000\) raise each year. Is this an example of linear or exponential growth? Find a function \(f\) that computes the salary during the \(n\) th year.

The table is a complete representation of \(f\). Use the table to determine if \(f\) is one-to-one and has an inverse. \(\begin{array}{rrrrrr}x & -2 & -1 & 0 & 1 & 2 \\ f(x) & 4 & 1 & 0 & 1 & 4\end{array}\)

Simplify the expression. $$ 7^{\log _{7} 2 x} $$

Comparing Growth Which function becomes larger for \(0 \leq x \leq 10: f(x)=2^{x}\) or \(g(x)=x^{2} ?\)

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=2 x-7 $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x^{2}-1, \quad g(x)=|x+1| $$

Near New Guinea there is a relationship between the number of bird species found on an island and the size of the island. The table lists the number of species of birds \(y\) found on an island with an area of \(x\) square kilometers. $$\begin{array}{rccccc}x\left(\mathrm{km}^{2}\right) & 0.1 & 1 & 10 & 100 & 1000 \\ \hline y \text { (species) } & 10 & 15 & 20 & 25 & 30\end{array}$$ (a) Find a function \(f\) that models the data. (b) Predict the number of bird species on an island of 5000 square kilometers. (c) Did your answer involve interpolation or extrapolation?

Simplify the expression. $$ 6^{\log _{5}(x+1)} $$

Comparing Growth Which function becomes larger for \(0 \leq x \leq 10: f(x)=4+3 x\) or \(g(x)=4(3)^{x} ?\)

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{2}-1 $$

Between 1950 and 1980 the use of chemical fertilizers increased. The table lists worldwide average usage \(y\) in kilograms per hectare of cropland, \(x\) years after \(1950 .\) (Note: 1 hectare \(\approx 2.47\) acres.) $$\begin{array}{ccccc}x & 0 & 13 & 22 & 29 \\ \hline y & 12.4 & 27.9 & 54.3 & 77.1\end{array}$$ (a) Graph the data. Are the data linear? (b) Find a function \(f\) that models the data. (c) Predict fertilizer usage in \(1989 .\) The actual value was 98.7 kilograms per hectare. What does this indicate about usage of fertilizer during the 1980 s?

Simplify the expression. $$ \log _{1 / 3}\left(\frac{1}{3}\right)^{64} $$

Comparing Growth Which function becomes larger for \(0 \leq x \leq 10: f(x)=2 x+1\) or \(g(x)=2^{-x} ?\)