Chapter 3

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(f(x)=8 x-x^{2}\) (a) \([-5,5]\) by \([-5,5]\) (b) \([-10,10]\) by \([-10,10]\) (c) \([-2,10]\) by \([-5,20]\) (d) \([-10,10]\) by \([-100,100]\)

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{2 / 5} $$

Evaluate the piecewise defined function at the indicated values. $$ \begin{array}{ll}{f(x)=\left\\{\begin{array}{ll}{x^{2}+2 x} & {\text { if } x \leq-1} \\ {x} & {\text { if }-11}\end{array}\right.} \\ {f(-4), f\left(-\frac{3}{2}\right), f(-1), f(0), f(25)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=x^{5} ; \quad g(x)=\sqrt[5]{x} $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=-|x| $$

23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=4-x^{2 / 3} $$

Evaluate the piecewise defined function at the indicated values. $$ \begin{array}{ll}{f(x)=\left\\{\begin{array}{ll}{3 x} & {\text { if } x<0} \\\ {x+1} & {\text { if } 0 \leq x \leq 2} \\ {(x-2)^{2}} & {\text { if } x>2} \\\ {(x-2)^{2}} & {\text { if } x>2}\end{array}\right.} \\ {f(-5), f(0), f(1), f(2), f(5)}\end{array} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ \begin{array}{l}{f(x)=x^{2}-4, \quad x \geq 0} \\ {g(x)=\sqrt{x+4}, \quad x \geq-4}\end{array} $$

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\sqrt[4]{-x} $$

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(h(x)=x^{3}-5 x-4\) (a) \([-2,2]\) by \([-2,2]\) (b) \([-3,3]\) by \([-10,10]\) (c) \([-3,3]\) by \([-10,5]\) (d) \([-10,10]\) by \([-10,10]\)

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=x^{2}+1 ; \quad f(x+2), f(x)+f(2) $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=x^{3}+1 ; \quad g(x)=(x-1)^{1 / 3} $$

Linear Functions Have Constant Rate of Change If \(f(x)=m x+b\) is a linear function, then the average rate of change of \(f\) between any two real numbers \(x_{1}\) and \(x_{2}\) is average rate of change \(=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}\) Calculate this average rate of change to show that it is the same as the slope \(m .\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\sqrt[3]{-x} $$

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(k(x)=\frac{1}{32} x^{4}-x^{2}+2\) (a) \([-1,1]\) by \([-1,1]\) (b) \([-2,2]\) by \([-2,2]\) (c) \([-5,5]\) by \([-5,5]\) (d) \([-10,10]\) by \([-10,10]\)

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=3 x-1 ; \quad f(2 x), 2 f(x) $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=2 x+3, \quad g(x)=4 x-1 $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=\frac{1}{x-1}, \quad x \neq 1 ; \quad g(x)=\frac{1}{x}+1, \quad x \neq 0 $$

Functions with Constant Rate of Change Are Linear If the function \(f\) has the same average rate of change \(c\) between any two points, then for the points \(a\) and \(x\) we have $$ c=\frac{f(x)-f(a)}{x-a} $$ Rearrange this expression to show that $$ f(x)=c x+(f(a)-c a) $$ and conclude that \(f\) is a linear function.

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{4} x^{2} $$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x<2} \\ {1} & {\text { if } x \geq 2}\end{array}\right.\)

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=x+4 ; \quad f\left(x^{2}\right),(f(x))^{2} $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=6 x-5, g(x)=\frac{x}{2} $$

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

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=-5 \sqrt{x} $$

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=6 x-18 ; \quad f\left(\frac{x}{3}\right), \frac{f(x)}{3} $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{2}, \quad g(x)=x+1 $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=\frac{x+2}{x-2} ; \quad g(x)=\frac{2 x+2}{x-1} $$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{3} & {\text { if } x<2} \\ {x-1} & {\text { if } x \geq 2}\end{array}\right.\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=3|x| $$

\(35-42\) A function is given. (a) Find all 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. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimal places. $$ f(x)=x^{3}-x $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\) $$f(x)=3 x+2$$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{3}+2, \quad g(x)=\sqrt[3]{x} $$

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=\frac{x-5}{3 x+4}, \quad g(x)=\frac{5+4 x}{1-3 x} $$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{1-x} & {\text { if } x<-2} \\ {5} & {\text { if } x \geq-2}\end{array}\right.\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{2}|\boldsymbol{X}| $$

\(35-42\) A function is given. (a) Find all 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. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimal places. $$ f(x)=3+x+x^{2}-x^{3} $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\) $$ f(x)=x^{2}+1 $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{1}{x}, g(x)=2 x+4 $$

Find the inverse function of \(f\) $$ f(x)=2 x+1 $$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{x} & {\text { if } x \leq 0} \\ {x+1} & {\text { if } x>0}\end{array}\right.\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=(x-3)^{2}+5 $$

\(35-42\) A function is given. (a) Find all 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. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimal places. $$ g(x)=x^{4}-2 x^{3}-11 x^{2} $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\) $$f(x)=5$$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{2}, \quad g(x)=\sqrt{x-3} $$

Find the inverse function of \(f\) $$ f(x)=6-x $$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{2 x+3} & {\text { if } x<-1} \\ {3-x} & {\text { if } x \geq-1}\end{array}\right.\)

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\sqrt{x+4}-3 $$

\(35-42\) A function is given. (a) Find all 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. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimal places. $$ g(x)=x^{5}-8 x^{3}+20 x $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\) $$ f(x)=\frac{1}{x+1} $$