91Ó°ÊÓ

Identify any interepts and test for symmetry. Then sketch the graph of the equation. \(y=\sqrt{1-x}\)

Decomposing a composite Function, find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are many correct answers.) $$h(x)=\frac{-x^{2}+3}{4-x^{2}}$$

Maximum Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volumes \(V\) (in cubic centimeters) of the box for various heights \(x\) (in centimeters). Use the table to estimate the maximum volume. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Height, } & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline \text { Volume, } V & {484} & {800} & {972} & {1024} & {980} & {864} \\ \hline\end{array} $$ (b) Plot the points \((x, V)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(V\) as a function of \(x ?\) (c) Given that \(V\) is a function of \(x,\) write the function and determine its domain.

Salary You are a sales representative for a clothing manufacturer. You are paid an annual salary, plus a bonus of 3\(\%\) of your sales over \(\$ 500,000 .\) Consider the two functions \(f(x)=x-500,000\) and \(g(x)=0.03 x\) When \(x\) is greater than \(\$ 500,000,\) which of the following represents your bonus? Explain your reasoning. $$\begin{array}{l}{\text { (a) } f(g(x))} \\ {\text { (b) } g(f(x))}\end{array}$$

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. The graph of \(y=-f(x)\) is a reflection of the graph of\(y=f(x)\) in the \(y\) -axis.

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. If the graph of the parent function \(f(x)=x^{2}\) is shifted six units to the right, three units up, and reflected in the \(x\) -axis, then the point \((-2,19)\) will lie on the graph of the transformation.

Describing Profits Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function \(f\) shown. The actual profits are shown by the function \(g\) along with a verbal description. Use the concepts of transformations of graphs to write \(g\) in terms of \(f .\) (a) The profits were only three-fourths as large as expected. (b) The profits were consistently \(\$ 10,000\) greater than predicted. (c) There was a two-year delay in the introduction of the product. After sales began, profits grew as expected.

Finding Parallel and Perpendicular, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. $$x-y=4, \quad(2.5,6.8)$$

Finding Parallel and Perpendicular, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. $$6 x+2 y=9, \quad(-3.9,-1.4)$$

Restricting the Domain In Exercises \(73-82,\) restrict the domain of the function \(f\) so that the function is one-to-one and has an inverse function. Then find the inverse function \(f^{-1} .\) State the domains and ranges of \(f\) and \(f^{-1} .\) Explain your results. (There are many correct answers.) $$f(x)=\frac{1}{2} x^{2}-1$$