Chapter 1

The table shows the revenue \(y\) (in thousands of dollars) of a landscaping business for each month of 2015, with \(x=1\) representing January. $$\begin{array}{|c|c|} \hline \text { Month, \(x\) } & \text { Revenue, \(y\) } \\ \hline 1 & 5.2 \\ 2 & 5.6 \\ 3 & 6.6 \\ 4 & 8.3 \\ 5 & 11.5 \\ 6 & 15.8 \\ 7 & 12.8 \\ 8 & 10.1 \\ 9 & 8.6 \\ 10 & 6.9 \\ 11 & 4.5 \\ 12 & 2.7 \\ \hline \end{array}$$ The mathematical model below represents the data. $$f(x)=\left\\{\begin{array}{l}-1.97 x+26.3 \\ 0.505 x^{2}-1.47 x+6.3\end{array}\right.$$ (a) Identify the independent and dependent variables and explain what they represent in the context of the problem. (b) What is the domain of each part of the piecewise-defined function? Explain your reasoning. (c) Use the mathematical model to find \(f(5).\) Interpret your result in the context of the problem. (d) Use the mathematical model to find \(f(11).\) Interpret your result in the context of the problem. (e) How do the values obtained from the models in parts (c) and (d) compare with the actual data values?

Compare the graph of \(g(x)=a x^{2}\) with the graph of \(f(x)=x^{2}\) when (a) \(01\).

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(t)=t^{2}+2 t-3$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=\frac{4}{x^{3}}$$

The research and development department of an automobile manufacturer has determined that when required to stop quickly to avoid an accident, the distance (in feet) a car travels during the driver's reaction time is given by \(R(x)=\frac{3}{4} x\) where \(x\) is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by \(B(x)=\frac{1}{15} x^{2}.\) (a) Find the function that represents the total stopping distance \(T\). (b) Use a graphing utility to graph the functions \(R, B\) and \(T\) in the same viewing window for \(0 \leq x \leq 60\). (c) Which function contributes most to the magnitude of the sum at higher speeds? Explain.

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(x)=x^{6}-2 x^{2}+3$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=\frac{6}{\sqrt{x}}$$

The force \(F\) (in tons) of water against the face of a dam is estimated by the function $$F(y)=149.76 \sqrt{10} y^{5 / 2}$$ where \(y\) is the depth of the water (in feet). (a) Complete the table. What can you conclude? $$\begin{array}{|l|l|l|l|l|l|} \hline y & 5 & 10 & 20 & 30 & 40 \\ \hline F(y) & & & & & \\ \hline \end{array}$$ (b) Use a graphing utility to graph the function. Describe your viewing window. (c) Use the table to approximate the depth at which the force against the dam is 1,000,000 tons. Verify your answer graphically. How could you find a better estimate?

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither.$$\begin{aligned}&L_{1}:(-2,-2),(2,10)\\\&L_{2}:(-1,3),(3,9)\end{aligned}$$

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$g(x)=x^{3}-5 x$$

The height \(y\) (in feet) of a baseball thrown by a child is $$y=-0.1 x^{2}+3 x+6$$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the glove of another child 30 feet away trying to catch the ball? Explain. (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results. (a) \(y=4 x\) (b) \(y=-4 x\) (c) \(y=\frac{1}{4} x\)

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither.$$\begin{aligned}&L_{1}:(-1,-7),(4,3)\\\&L_{2}:(1,5),(-2,-7)\end{aligned}$$.

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$h(x)=x^{5}-4 x^{3}$$

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results. (a) \(y=\frac{2}{3} x\) (b) \(y=-\frac{3}{2} x\) (c) \(y=\frac{2}{3} x+2\)

Find the domain of the function.$$f(x)=\frac{4}{9-x}$$

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(x)=x \sqrt{1-x^{2}}$$

A company owns two retail stores. The annual sales (in thousands of dollars) of the stores each year from 2009 through 2015 can be approximated by the models $$S_{1}=973+1.3 t^{2} \quad \text { and } \quad S_{2}=349+72.4 t$$ where \(t\) is the year, with \(t=9\) corresponding to 2009. (a) Write a function \(T\) that represents the total annual sales of the two stores. (b) Use a graphing utility to graph \(S_{1}, S_{2},\) and \(T\) in the same viewing window.

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outermost ripple is given by \(r(t)=0.6 t,\) where \(t\) is the time (in seconds) after the pebble strikes the water. The area of the circle is given by \(A(r)=\pi r t^{2} .\) Find and interpret \((A \circ r)(t).\)

Find the difference quotient and simplify your answer. $$f(x)=2 x, \quad \frac{f(x+c)-f(x)}{c}, \quad c \neq 0$$

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results. (a) \(y=-\frac{1}{2} x\) (b) \(y=-\frac{1}{2} x+3\) (c) \(y=2 x-4\)

Find the domain of the function.$$f(x)=\frac{\sqrt{x-5}}{x-7}$$

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(x)=x \sqrt{x+5}$$

Find the difference quotient and simplify your answer. $$g(x)=3 x-1, \frac{g(x+h)-g(x)}{h}, \quad h \neq 0$$

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results. (a) \(y=x-8\) (b) \(y=x+1\) (c) \(y=-x+3\)

Find the domain of the function.$$f(x)=\sqrt{100-x^{2}}$$.

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$g(s)=4 s^{2 / 3}$$

The number of bacteria in a refrigerated food product is given by $$N(T)=10 T^{2}-20 T+600, \quad 1 \leq T \leq 20$$ where \(T\) is the temperature of the food in degrees Celsius. When the food is removed from the refrigerator, the temperature of the food is given by $$T(t)=2 t+1$$ where \(t\) is the time in hours. (a) Find the composite function \(N(T(t))\) or \((N \circ T)(t)\) and interpret its meaning in the context of the situation. (b) Find \((N \circ T)(12)\) and interpret its meaning. (c) Find the time when the bacteria count reaches \(1200 .\)

Find the difference quotient and simplify your answer. $$f(x)=x^{2}-x+1, \quad \frac{f(2+h)-f(2)}{h}, \quad h \neq 0$$

Find the domain of the function.$$f(x)=\sqrt[3]{16-x^{2}}$$.

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(s)=4 s^{3 / 5}$$

The spread of a contaminant is increasing in a circular pattern on the surface of a lake. The radius of the contaminant can be modeled by \(r(t)=5.25 \sqrt{t},\) where \(r\) is the radius in meters and \(t\) is time in hours since contamination. (a) Find a function that gives the area \(A\) of the circular leak in terms of the time \(t\) since the spread began. (b) Find the size of the contaminated area after 36 hours. (c) Find when the size of the contaminated area is 6250 square meters.

Find the difference quotient and simplify your answer. $$f(x)=x^{3}+x, \quad \frac{f(x+h)-f(x)}{h}, \quad h \neq 0$$

Highway Engineering When driving down a mountain road, you notice warning signs indicating that it is a "12\% grade." This means that the slope of the road is \(-\frac{12}{100}\). Approximate the amount of horizontal change in your position if you note from elevation markers that you have descended 2000 feet vertically.

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=4-x$$

An air traffic controller spots two planes flying at the same altitude. Their flight paths form a right angle at point \(P\). One plane is 150 miles from point \(P\) and is moving at 450 miles per hour. The other plane is 200 miles from point \(P\) and is moving at 450 miles per hour. Write the distance \(s\) between the planes as a function of time \(t.\)

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=4 x+8$$

The suggested retail price of a new car is \(p\) dollars. The dealership advertised a factory rebate of \(\$ 2000\) and a \(9 \%\) discount. (a) Write a function \(R\) in terms of \(p\) giving the cost of the car after receiving the rebate from the factory. (b) Write a function \(S\) in terms of \(p\) giving the cost of the car after receiving the dealership discount. (c) Form the composite functions \((R \circ S)(p)\) and \((S \circ R)(p)\) and interpret each. (d) Find \((R \circ S)(24,795)\) and \((S \circ R)(24,795) .\) Which yields the lower cost for the car? Explain.

Determine whether the statement is true or false. Justify your answer. The set of ordered pairs \(\\{(-8,-2),(-6,0),(-4,0) (-2,2),(0,4),(2,-2)\\}\) represents a function.

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=x^{2}-9$$

You are given the dollar value of a product in 2015 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let \(t=15\) represent \(2015 .\) ) 2015 Value \(\$ 2540\) Rate \(\$ 125\) increase per year

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=x^{2}-4 x$$

Determine whether the statement is true or false. Justify your answer. Given two functions \(f\) and \(g,\) you can calculate \((f \circ g)(x)\) if and only if the range of \(g\) is a subset of the domain of \(f\).

You are given the dollar value of a product in 2015 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let \(t=15\) represent \(2015 .\) ) 2015 Value \(\$ 156\) Rate \(\$ 5.50\) increase per year

The cost of parking in a metered lot is \(\$ 1.00\) for the first hour and \(\$ 0.50\) for each additional hour or portion of an hour. (a) \(\mathrm{A}\) customer needs a model for the cost \(C\) of parking in the metered lot for \(t\) hours. Which of the following is the appropriate model? \(C_{1}(t)=1+0.50[t-1]\) \(C_{2}(t)=1-0.50[-(t-1)]\) (b) Use a graphing utility to graph the appropriate model. Estimate the cost of parking in the metered lot for 7 hours and 10 minutes.

The function in Example 9 can be decomposed in other ways. For which of the following pairs of functions is \(h(x)=\frac{1}{(x-2)^{2}}\) equal to \(f(g(x)) ?\) (a) \(g(x)=\frac{1}{x-2}\) and \(f(x)=x^{2}\) (b) \(g(x)=x^{2}\) and \(f(x)=\frac{1}{x-2}\) (c) \(g(x)=(x-2)^{2}\) and \(f(x)=\frac{1}{x}\)

Given \(f(x)=x^{2},\) is \(f\) the independent variable? Why or why not?

Business The cost of parking in a metered lot is \(\$ 1.00\) for the first hour and \(\$ 0.50\) for each additional hour or portion of an hour. (a) A customer needs a model for the cost \(C\) of parking in the metered lot for \(t\) hours. Which of the following is the appropriate model? \\[ \begin{array}{l} C_{1}(t)=1+0.50[t-1] \\ C_{2}(t)=1-0.50[-(t-1)] \end{array} \\] (b) Use a graphing utility to graph the appropriate model. Estimate the cost of parking in the metered lot for 7 hours and 10 minutes.

You are given the dollar value of a product in 2015 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let \(t=15\) represent \(2015 .\) ) 2015 Value \(\$ 20,400\) Rate \(\$ 2000\) decrease per year

The cost of sending an overnight package from New York to Atlanta is \(\$ 23.20\) for a package weighing up to but not including 1 pound and \(\$ 2.00\) for each additional pound or portion of a pound. Use the greatest integer function to create a model for the cost C of overnight delivery of a package weighing \(x\) pounds, where \(x>0 .\) Sketch the graph of the function.