Chapter 2

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

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 \(|f(x)|<|g(x)|\) $$|6 x+9|=|6 x-3|$$

Use the analytic 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 your calculator and the standard window to support your conclusion. $$f(x)=|-x|$$

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

For each function find \(f(x+h)\) and \(f(x)+f(h)\). $$f(x)=x^{2}-4$$

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 \(|f(x)|<|g(x)|\) $$|0.25 x+1|=|0.75 x-3|$$

Use the analytic 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 your calculator and the standard window to support your conclusion. $$f(x)=\frac{1}{4 x^{3}}$$

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

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

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 \(|f(x)|<|g(x)|\) $$|0.40 x+2|=|0.60 x-5|$$

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

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

Sales of Apple Products Average household spending on Apple products is shown in the figure for both U.S. sales and worldwide sales. Use this figure (Figure can't 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.

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.) $$y=10 \sqrt{x-20}+5$$

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

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 \(|f(x)|<|g(x)|\) $$|5 x-6|=|-5 x+6|$$

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.) $$y=-2 \sqrt{x+15}-18$$

Solve each equation graphically. $$|x+1|+|x-6|=11$$

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

The table lists the average annual costs (in dollars) of tuition and fees at public four-year colleges for selected years. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Tuition and Fees (in dollars) } \\ \hline 2000 & 3505 \\ 2003 & 4632 \\ 2005 & 5491 \\ 2008 & 6532 \\ 2010 & 7605 \end{array}$$ (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 2009 to the nearest hundred dollars.

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.) $$y=-0.5 \sqrt{x+10}+5$$

Solve each equation graphically. $$|2 x+2|+|x+1|=9$$

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

The table shows how the percent of women in the civilian workforce has changed from 1970 to 2010 $$\begin{array}{l|c} \text { Year } & \begin{array}{c} \text { Percent of Women in } \\ \text { the Workforce } \end{array} \\ \hline 1970 & 43.3 \\\ 1975 & 46.3 \\ 1980 & 51.5 \\ 1985 & 54.5 \\ 1990 & 57.5 \\ 1995 & 58.9 \\\ 2000 & 59.9 \\ 2005 & 59.0 \\ 2010 & 58.6 \end{array}$$ (a) Use a calculator to find the least-squares regression line for these data, where \(x\) is the number of years after 1970 (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 percent of population of women in the civilian workforce in 2015 .

Solve each equation graphically. $$|x|+|x-4|=8$$

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

Recall from Chapter 1 that a unique line is determined by two distinct points on the line and that the values of \(m\) and \(b\) can then be determined for the general form of the linear function $$ f(x)=m x+b $$ Sketch by hand the line that passes through the points \((1,-2)\) and \((3,2)\)

Solve each equation graphically. $$|0.5 x+2|+|0.25 x+4|=9$$

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

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-50| \leq 22,\) Boston, Massachusetts

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

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-10| \leq 36,\) Chesterfield, Canada

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

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-61.5| \leq 12.5,\) Buenos Aires, Argentina

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

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-43.5| \leq 8.5,\) Punta Arenas, Chile

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

Solve each problem. Dr. Cazayoux has found that, over the years, \(95 \%\) of the babies he delivered weighed \(x\) pounds, where \(|x-8.0| \leq 1.5 .\) What range of weights corresponds to this inequality?

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

Solve each problem. The industrial process that is used to convert methanol to gasoline is carried out at a temperature range of \(680^{\circ} \mathrm{F}\) to \(780^{\circ} \mathrm{F}\). Using \(F\) as the variable, write an absolute value inequality that corresponds to this range.

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

Solve each problem. Systolic blood pressure is the maximum pressure produced by each heartbeat. Both low blood pressure and high blood pressure are cause for medical concern. Therefore, health care professionals are interested in a patient's "pressure difference from normal," or \(P_{d}\). If 120 is considered a normal systolic pressure, \(P_{d}=|P-120|,\) where \(P\) is the patient's recorded systolic pressure. For example, a patient with a systolic pressure \(P\) of 113 would have a pressure difference from normal of \(P_{d}=|P-120|=|113-120|=|-7|=7\) (a) Calculate the \(P_{d}\) value for a woman whose actual systolic pressure is 116 and whose normal value should be 125 (b) If a patient's \(P_{d}\) value is 17 and the normal pressure for his sex and age should be \(120,\) what are the two possible values for his systolic blood pressure?

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

Solve each problem. When a model kite was flown in crosswinds in tests, it attained speeds of 98 to 148 feet per second in winds of 16 to 26 feet per second. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.

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

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=2 x+1,\) and the difference between \(y\) and 1 is less than 0.1.

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=3 x-6,\) and the difference between \(y\) and 2 is less than 0.01.

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=\left(11 x^{2}+12 x\right)^{2}$$

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=4 x-8,\) and the difference between \(y\) and 3 is less than 0.001.

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=\sqrt{x^{2}-1}$$