Chapter 1

Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function.

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 \(\$ 245,000\) Rate \(\$ 5600\) decrease per year

Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.

Perform the operation and simplify. $$12-\frac{4}{x+2}$$

Accounting A school district purchases a high-volume printer, copier, and scanner for \(\$ 25,000\). After 10 years, the equipment will have to be replaced. Its value at that time is expected to be \(\$ 2000\). (a) Write a linear equation giving the value \(V\) of the equipment for each year \(t\) during its 10 years of use. (b) Use a graphing utility to graph the linear equation representing the depreciation of the equipment, and use the value or trace feature to complete the table. Verify your answers algebraically by using the equation you found in part (a). $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline V & & & & & & & & & & & \\ \hline \end{array}$$

Three siblings are of three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one-half the age of the youngest. (a) Write a composite function that gives the oldest sibling's age in terms of the youngest. Explain how you arrived at your answer. (b) The oldest sibling is 16 years old. Find the ages of the other two siblings.

Perform the operation and simplify. $$\frac{3}{x^{2}+x-20}+\frac{2 x}{x^{2}+4 x-5}$$

Meterology Recall that water freezes at \(0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)\) and boils at \(100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)\). (a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius \(C\) and degrees Fahrenheit \(F\). (b) Use the result of part (a) to complete the table. $$\begin{array}{|c|c|c|c|c|c|c|} \hline C & & -10^{\circ} & 10^{\circ} & & & 177^{\circ} \\ \hline F & 0^{\circ} & & & 68^{\circ} & 90^{\circ} & \\ \hline \end{array}$$

The number \(N\) (in thousands) of existing condominiums and cooperative homes sold each year from 2010 through 2013 in the United States is approximated by the model $$\begin{array}{l} N=-24.83 t^{3}+906 t^{2}-10,928.2 t+44,114 \\ 10 \leq t \leq 13 \end{array}$$ where \(t\) represents the year, with \(t=10\) corresponding to 2010 . (a) Use a graphing utility to graph the model over the appropriate domain. (b) Use the graph from part (a) to determine during which years the number of cooperative homes and condos was increasing. During which years was the number decreasing? (c) Approximate the minimum number of cooperative homes and condos sold from 2010 through 2013 .

Three siblings are of three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one-half the age of the youngest. (a) Write a composite function that gives the youngest sibling's age in terms of the oldest. Explain how you arrived at your answer. (b) The youngest sibling is two years old. Find the ages of the other two siblings.

Perform the operation and simplify. $$\frac{x^{5}}{2 x^{3}+4 x^{2}} \cdot \frac{4 x+8}{3 x}$$

Business \(A\) contractor purchases a bulldozer for \(\$ 36,500 .\) The bulldozer requires an average expenditure of \(\$ 11.25\) per hour for fuel and maintenance, and the operator is paid \(\$ 19.50\) per hour. (a) Write a linear equation giving the total cost \(C\) of operating the bulldozer for \(t\) hours. (Include the purchase cost of the bulldozer.) (b) Assuming that customers are charged \(\$ 80\) per hour of bulldozer use, write an equation for the revenue \(R\) derived from \(t\) hours of use. (c) Use the profit formula \((P=R-C)\) to write an equation for the profit gained from \(t\) hours of use. (d) Use the result of part (c) to find the break-even point (the number of hours the bulldozer must be used to gain a profit of 0 dollars).

Perform the operation and simplify. $$\frac{x+7}{2(x-9)} \div \frac{x-7}{2(x-9)}$$

Real Estate \(\quad\) A real estate office handles an apartment complex with 50 units. When the rent per unit is \(\$ 580\) per month, all 50 units are occupied. However, when the rent is \(\$ 625\) per month, the average number of occupied units drops to \(47 .\) Assume that the relationship between the monthly rent \(p\) and the demand \(x\) is linear. (a) Write an equation of the line giving the demand \(x\) in terms of the rent \(p\) (b) Use a graphing utility to graph the demand equation and use the trace feature to estimate the number of units occupied when the rent is \(\$ 655 .\) Verify your answer algebraically. (c) Use the demand equation to predict the number of units occupied when the rent is lowered to \(\$ 595 .\) Verify your answer graphically.

Determine whether the statement is true or false. Justify your answer. A function with a square root cannot have a domain that is the set of all real numbers.

Find three points that lie on the graph of the equation. (There are many correct answers.) $$y=-x^{2}+x-5$$

In 1994 , Penn State University had an enrollment of 73,500 students. By \(2013,\) the enrollment had increased to \(98,097 . (a) What was the average annual change in enrollment from 1994 to \)2013 ?\( (b) Use the average annual change in enrollment to estimate the enrollments in \)1996,2006,$ and 2011 . (c) Write an equation of a line that represents the given data. What is its slope? Interpret the slope in the context of the problem.

Determine whether the statement is true or false. Justify your answer. It is possible for an odd function to have the interval \([0, \infty)\) as its domain.

Find three points that lie on the graph of the equation. (There are many correct answers.) $$y=\frac{1}{5} x^{3}-4 x^{2}+1$$

Find three points that lie on the graph of the equation. (There are many correct answers.) $$x^{2}+y^{2}=49$$

Determine whether the statement is true or false. Justify your answer. The line through (-8,2) and (-1,4) and the line through (0,-4) and (-7,7) are parallel.

Find three points that lie on the graph of the equation. (There are many correct answers.) $$y=\frac{x}{x^{2}-5}$$

Determine whether the statement is true or false. Justify your answer. If the points (10,-3) and (2,-9) lie on the same line, then the point \(\left(-12,-\frac{37}{2}\right)\) also lies on that line.

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=x^{3}+x+1$$

Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture about what \(a\) and \(b\) represent. Verify your conjecture. $$\frac{x}{7}+\frac{y}{-3}=1$$

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=x \sqrt{4-x^{2}}$$

Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture about what \(a\) and \(b\) represent. Verify your conjecture. $$\frac{x}{-6}+\frac{y}{2}=1$$

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=\frac{3 x^{2}}{x^{2}+1}$$

Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture about what \(a\) and \(b\) represent. Verify your conjecture. $$\frac{x}{4}+\frac{y}{-\frac{2}{3}}=1$$

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=\frac{4 x}{\sqrt{x^{2}+15}}$$

Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture about what \(a\) and \(b\) represent. Verify your conjecture. $$\frac{x}{\frac{1}{2}}+\frac{y}{5}=1$$

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(f^{-1} \circ g^{-1}\right)(1)$$

Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: (2,0) \(y\) -intercept: (0,9)

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(g^{-1} \circ f^{-1}\right)(-3)$$

Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: (-5,0) \(y\) -intercept: (0,-4)

Can you represent the greatest integer function using a piecewise-defined function?

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(f^{-1} \circ f^{-1}\right)(-6)$$

Think About It Can you represent the greatest integer function using a piecewise-defined function?

How does the graph of the greatest integer function differ from the graph of a line with a slope of zero?

Think About It How does the graph of the greatest integer function differ from the graph of a line with a slope of zero?

Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: \(\left(\frac{3}{4}, 0\right)\) \(y\) -intercept: \(\left(0, \frac{4}{3}\right)\)

Let \(f\) be an even function. Determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=-f(x+3)\)

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$f^{-1} \circ g^{-1}$$

Think About It Let \(f\) be an even function. Determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=-f(x+3)\)

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$g^{-1} \circ f^{-1}$$

Prove that a function of the following form is odd. $$y=a_{2 n+1} x^{2 n+1}+a_{2 n-1} x^{2 n-1}+\cdots+a_{3} x^{3}+a_{1} x$$

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$g^{-1} \circ f^{-1}$$

Prove that a function of the following form is even. $$y=a_{2 n} x^{2 n}+a_{2 n-2} x^{2 n-2}+\cdots+a_{2} x^{2}+a_{0}$$

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$f^{-1} \circ g^{-1}$$

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$-2 x^{2}+11 x+3$$