Chapter 2

Solve each problem. Organic Food Sales Organic food sales in the United States in millions of dollars \(x\) years past 2005 can be modeled by \(O(x)=2649.4 x+13,260\) (a) Evaluate \(O(9)\) and interpret your result. (b) Use the formula for \(O(x)\) to write an equation that gives the organic food sales \(y\) during year \(x\). (c) Refer to part (b) and find \(y\) when \(x=2014\). (d) Use your equation in part (b) to determine the year when organic food sales reached \(\$ 26,507\) million.

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$-f(x)$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{x^{2}+5}{x}$$

For each function find (a) \(f(x+h)\) and (b) \(f(x)+f(h)\) $$f(x)=5 x^{2}+x$$

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$f(x-3)+1$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\sqrt[3]{x^{3}-5 x}$$+

For each function find (a) \(f(x+h)\) and (b) \(f(x)+f(h)\) $$f(x)=3 x-x^{2}$$

Sales of Apple Products Average household spending on Apple products is shown in the figure for both \(U . S .\) sales and worldwide sales that exclude U.S. sales. Use this figure for Exercises 73-74. (Image cannot copy) U.S. sales in dollars can be approximated during year \(x\) by $$U(x)=13(x-2006)^{2}+115$$ Evaluate \(U(2011)\) and interpret your result.

Solve each equation or inequality. $$\left|6-\frac{1}{3} x\right|>0$$

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$f(2 x)$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x$$

For each function find (a) \(f(x+h)\) and (b) \(f(x)+f(h)\) $$f(x)=x^{3}$$

Solve each equation or inequality. $$|8 x-4|<0$$

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$2 f(x-1)$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{x^{2}+3}{|x|}$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=4 x+3$$

Cost of Public College Education The table lists the average annual costs (in dollars) of tuition and fees at public four-year colleges for selected years. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Tuition and Fees (in dollars) } \\ \hline 2000 & 3505 \\ 2005 & 5491 \\ 2010 & 7605 \\ 2015 & 9420 \\ \hline \end{array}$$ (a) Use a calculator to find the least-squares regression line for these data, where \(x\) is the number of years after 2000 (b) Based on your result from part (a), write an equation that yields the same \(y\) -values when the actual year is entered. (c) Estimate the cost of tuition and fees in 2014 to the nearest hundred dollars.

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$3 f\left(\frac{1}{4} x\right)$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=9$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=5 x-6$$

Video-on-Demand The following table shows the projected revenue earned in various years by the U.S. "Video-On-Demand" market segment in millions of dollars. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Revenue (in 5 millions) } \\ \hline 2015 & 9040 \\ 2016 & 9529 \\ 2017 & 10,000 \\ 2018 & 10,436 \\ 2019 & 10,825 \\ 2020 & 11,162 \\ 2021 & 11,448 \\ \hline \end{array}$$ (a) Use a calculator to find the least-squares regression line for these data, where \(x\) is the number of years after 2015 (b) Based on your result from part (a), write an equation that yields the same \(y\) -values when the actual year is entered. (c) Predict the revenue for this market segment to the nearest million dollars in 2025 .

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$-2 f(4 x)$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=-x^{3}+2 x$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=-6 x^{2}-x+4$$

Sketch by hand the line that passes through the points \((1,-2)\) and \((3,2)\).

Solve each equation or inequality. $$|7 x-5| \geq-5$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=x^{5}-2 x^{3}$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\frac{1}{2} x^{2}+4 x$$

Explain how to solve an equation of the form \(|a x+b|=|c x+d|\) analytically.

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$-2 f(-x)$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=0.5 x^{4}-2 x^{2}+1$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=x^{3}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|3 x+1|=|2 x-7|$$

Let the domain of \(f(x)\) be [-1,2] and the range be \([0,3] .\) Find the domain and range of the following. $$f(-3 x)$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=0.75 x^{2}+|x|+1$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=-2 x^{3}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|x-4|=|7 x+12|$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=1-x^{2}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|-2 x+5|=|x+3|$$

each function has a graph with an endpoint (a translation of the point (0,0) .) Enter each into your calculator in an appropriate viewing window, and, using your knowledge of the graph of \(y=\sqrt{x}\), determine the domain and range of the function. (Hint: Locate the endpoint.) $$y=10 \sqrt{x-20}+5$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=x^{2}+2 x$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|-5 x+1|=|3 x-4|$$

each function has a graph with an endpoint (a translation of the point (0, 0).) Enter each into your calculator in an appropriate viewing window, and, using your knowledge of the graph of \(y=\sqrt{x},\) determine the domain and range of the function. (Hint: Locate the endpoint.) $$y=-2 \sqrt{x+15}-18$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=x^{6}-4 x^{3}$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=3 x^{2}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$\left|x-\frac{1}{2}\right|=\left|\frac{1}{2} x-2\right|$$

In Exercises \(81-83 \text { , each function has a graph with an endpoint (a translation of the point }(0,0) .)\) Enter each into your calculator in an appropriate viewing window, and, using your knowledge of the graph of \(y=\sqrt{x}\), determine the domain and range of the function. (Hint: Locate the endpoint.) $$y=-0.5 \sqrt{x+10}+5$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=x^{3}-3 x$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\sqrt{x}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|x+3|=\left|\frac{1}{3} x+8\right|$$