Chapter 3

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

Find the inverse function of \(f\) $$ f(x)=4 x+7 $$

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

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=3-\frac{1}{2}(x-1)^{2} $$

\(35-42\) 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. $$ U(x)=x \sqrt{6-x} $$

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

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

Find the inverse function of \(f\) $$ f(x)=3-5 x $$

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

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

\(35-42\) 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. $$ U(x)=x \sqrt{x-x^{2}} $$

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

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

Find the inverse function of \(f\) $$ f(x)=5-4 x^{3} $$

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

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

\(35-42\) 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. $$ V(x)=\frac{1-x^{2}}{x^{3}} $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\) $$ f(x)=3-5 x+4 x^{2} $$

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

Find the inverse function of \(f\) $$ f(x)=\frac{1}{x^{2}}, \quad x>0 $$

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

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

\(35-42\) 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. $$ V(x)=\frac{1}{x^{2}+x+1} $$

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

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

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

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

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{2} \sqrt{x+4}-3 $$

Find the domain of the function. $$ f(x)=2 x $$

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

Find the inverse function of \(f\) $$ f(x)=\frac{x-2}{x+2} $$

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

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=3-2(x-1)^{2} $$

Find the domain of the function. $$ f(x)=x^{2}+1 $$

Find \(f \circ g \circ h\) $$ f(x)=x-1, \quad g(x)=\sqrt{x}, \quad h(x)=x-1 $$

Find the inverse function of \(f\) $$ f(x)=\frac{x}{x+4} $$

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

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{2} ;\) shift upward 3 units

Find the domain of the function. $$ f(x)=2 x, \quad-1 \leq x \leq 5 $$

Find the inverse function of \(f\) $$ f(x)=\frac{3 x}{x-2} $$

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{-x} & {\text { if } x \leq 0} \\ {9-x^{2}} & {\text { if } 0< x \leq 3} \\ {x-3} & {\text { if } x>3}\end{array}\right.\)

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{3} ;\) shift downward 1 unit

Find the domain of the function. $$ f(x)=x^{2}+1, \quad 0 \leq x \leq 5 $$

Find \(f \circ g \circ h\) $$ f(x)=x^{4}+1, \quad g(x)=x-5, \quad h(x)=\sqrt{x} $$

Find the inverse function of \(f\) $$ f(x)=\frac{2 x+5}{x-7} $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt{x},\) shift 2 units to the left

Find the domain of the function. $$f(x)=\frac{1}{x-3}$$

Find \(f \circ g \circ h\) $$ f(x)=\sqrt{x}, \quad g(x)=\frac{x}{x-1}, \quad h(x)=\sqrt[3]{x} $$

Find the inverse function of \(f\) $$ f(x)=\frac{4 x-2}{3 x+1} $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt[3]{x},\) shift 1 unit to the right