Chapter 3

Find the inverse function of \(f\). \(c(x)=4+\sqrt[3]{x}\)

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

\(45-50\) Express the function in the form \(f \circ g\) $$ F(x)=\sqrt{x}+1 $$

\(45-46=\) A quadratic function is given. (a) Use a graphing device to find the maximum or minimum value of the quadratic function \(f,\) correct to two decimal places. (b) Find the exact maximum or minimum value of \(f,\) and compare with your answer to part (a). $$ f(x)=1+x-\sqrt{2} x^{2} $$

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{1-x^{2}} & {\text { if } x \leq 2} \\ {x} & {\text { if } x>2}\end{array}\right. $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=|x-1| $$

Find the inverse function of \(f\). \(f(x)=\left(2-x^{3}\right)^{5}\)

Find the domain of the function. $$ f(x)=\sqrt[4]{x+9} $$

\(45-50\) Express the function in the form \(f \circ g\) $$ G(x)=\frac{x^{2}}{x^{2}+4} $$

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{0} & {\text { if }|x| \leq 2} \\ {3} & {\text { if }|x|>2}\end{array}\right. $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=|x+2|+2 $$

Find the inverse function of \(f\). \(f(x)=1+\sqrt{1+x}\)

Find the domain of the function. $$ f(t)=\sqrt[3]{t-1} $$

\(45-50\) Express the function in the form \(f \circ g\) $$ G(x)=\frac{1}{x+3} $$

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{x^{2}} & {\text { if }|x| \leq 1} \\ {1} & {\text { if }|x|>1}\end{array}\right. $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=2-|x| $$

Find the domain of the function. $$ g(x)=\sqrt{7-3 x} $$

\(45-50\) Express the function in the form \(f \circ g\) $$ H(x)=\left|1-x^{3}\right| $$

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{4} & {\text { if } x<-2} \\ {x^{2}} & {\text { if }-2 \leq x \leq 2} \\ {-x+6} & {\text { if } x>2}\end{array}\right. $$

49–52 ? Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8,8]\) by \([-2,8]\) $$ \begin{array}{ll}{\text { (a) } y=\sqrt[4]{x}} & {\text { (b) } y=\sqrt[4]{x+5}} \\ {\text { (c) } y=2 \sqrt[4]{x+5}} & {\text { (d) } y=4+2 \sqrt[4]{x+5}}\end{array} $$

Find the inverse function of \(f\). \(f(x)=x^{4}, \quad x \geq 0\)

Find the domain of the function. $$ h(x)=\sqrt{2 x-5} $$

\(45-50\) Express the function in the form \(f \circ g\) $$ H(x)=\sqrt{1+\sqrt{x}} $$

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{-x} & {\text { if } x \leq 0} \\ {9-x^{2}} & {\text { if } 03}\end{array}\right. $$

49–52 ? Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8,8]\) by \([-6,6]\) $$ \begin{array}{ll}{\text { (a) } y=|x|} & {\text { (b) } y=-|x|} \\ {\text { (c) } y=-3|x|} & {\text { (d) } y=-3|x-5|}\end{array} $$

Find the inverse function of \(f\). \(f(x)=1-x^{3}\)

Find the domain of the function. $$ G(x)=\sqrt{x^{2}-9} $$

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ f(x)=x^{3}-x $$

49–52 ? Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-4,6]\) by \([-4,4]\) $$ \begin{array}{ll}{\text { (a) } y=x^{6}} & {\text { (b) } y=\frac{1}{3} x^{6}} \\\ {\text { (c) } y=-\frac{1}{3} x^{6}} & {\text { (d) } y=-\frac{1}{3}(x-4)^{6}}\end{array} $$

A function \(f\) is given. (a) Sketch the graph of \(f\) (b) Use the graph of \(f\) to sketch the graph of \(f^{-1} .\) (c) Find \(f^{-1} .\) \(f(x)=3 x-6\)

Find the domain of the function. $$ g(x)=\frac{\sqrt{2+x}}{3-x} $$

\(51-54\) Express the function in the form \(f \circ g \circ h\) $$ F(x)=\sqrt[3]{\sqrt{x}-1} $$

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ f(x)=3+x+x^{2}-x^{3} $$

A function \(f\) is given. (a) Sketch the graph of \(f\) (b) Use the graph of \(f\) to sketch the graph of \(f^{-1} .\) (c) Find \(f^{-1} .\) \(f(x)=16-x^{2}, \quad x \geq 0\)

Find the domain of the function. $$ g(x)=\frac{\sqrt{x}}{2 x^{2}+x-1} $$

\(51-54\) Express the function in the form \(f \circ g \circ h\) $$ G(x)=(4+\sqrt[3]{x})^{9} $$

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ g(x)=x^{4}-2 x^{3}-11 x^{2} $$

A function \(f\) is given. (a) Sketch the graph of \(f\) (b) Use the graph of \(f\) to sketch the graph of \(f^{-1} .\) (c) Find \(f^{-1} .\) \(f(x)=\sqrt{x+1}\)

Find the domain of the function. $$ g(x)=\sqrt[4]{x^{2}-6 x} $$

\(51-54\) Express the function in the form \(f \circ g \circ h\) $$ G(x)=\frac{2}{(3+\sqrt{x})^{2}} $$

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ g(x)=x^{5}-8 x^{3}+20 x $$

A function \(f\) is given. (a) Sketch the graph of \(f\) (b) Use the graph of \(f\) to sketch the graph of \(f^{-1} .\) (c) Find \(f^{-1} .\) \(f(x)=x^{3}-1\)

Find the domain of the function. $$ g(x)=\sqrt{x^{2}-2 x-8} $$

\(55-56\) : Revenue, Cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that revenue \(=\) price per item \(\times\) number of items sold to express \(R(x),\) the revenue from an order of \(x\) stickers, as a product of two functions of \(x .\)

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ U(x)=x \sqrt{6-x} $$

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=x^{3}-x\)

Find the domain of the function. $$ f(x)=\frac{3}{\sqrt{x-4}} $$

\(55-56\) : Revenue, Cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that profit \(=\) revenue \(-\) cost to express \(P(x),\) the profit on an order of \(x\) stickers, as a difference of two functions of \(x .\)

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ U(x)=x \sqrt{x-x^{2}} $$

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=x^{3}+x\)