Chapter 3

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. $$ f(x)=x^{3}+2 $$

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

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

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{1}{\sqrt{x}}, \quad g(x)=x^{2}-4 x $$

\(29-38=\) Find the maximum or minimum value of the function. $$ g(x)=2 x(x-4)+7 $$

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

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. $$ f(x)=-x^{3} $$

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

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

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\sqrt[3]{x}, \quad g(x)=\sqrt[4]{x} $$

Find a function whose graph is a parabola with vertex \((1,-2)\) and that passes through the point \((4,16) .\)

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

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. $$ y=1+\sqrt{x} $$

Find the inverse function of \(f\). \(f(x)=\frac{1+3 x}{5-2 x}\)

Functions That Are Always Increasing or Decreasing Sketch rough graphs of functions that are defined for all real numbers and that exhibit the indicated behavior (or explain why the behavior is impossible). (a) \(f\) is always increasing, and \(f(x)>0\) for all \(x\) (b) \(f\) is always decreasing, and \(f(x)>0\) for all \(x\) (c) \(f\) is always increasing, and \(f(x)<0\) for all \(x\) (d) \(f\) is always decreasing, and \(f(x)<0\) for all \(x\)

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

\(29-40\) 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 a function whose graph is a parabola with vertex \((3,4)\) and that passes through the point \((1,-8) .\)

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

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. $$ y=2-\sqrt{x+1} $$

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

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

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=x-1, \quad g(x)=\sqrt{x}, \quad h(x)=x-1 $$

\(41-44=\) Find the domain and range of the function. $$ f(x)=-x^{2}+4 x-3 $$

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

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. $$ y=\frac{1}{2} \sqrt{x+4}-3 $$

Find the inverse function of \(f\). \(f(x)=\sqrt{2+5 x}\)

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

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=\frac{1}{x}, \quad g(x)=x^{3}, \quad h(x)=x^{2}+2 $$

\(41-44=\) Find the domain and range of the function. $$ f(x)=x^{2}-2 x-3 $$

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

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. $$ y=3-2(x-1)^{2} $$

Find the inverse function of \(f\). \(f(x)=x^{2}+x, \quad x \geq-\frac{1}{2}\)

Find the domain of the function. $$ f(x)=\frac{1}{3 x-6} $$

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=x^{4}+1, \quad g(x)=x-5, \quad h(x)=\sqrt{x} $$

\(41-44=\) Find the domain and range of the function. $$ f(x)=2 x^{2}+6 x-7 $$

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

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. $$ y=5+(x+3)^{2} $$

Find the inverse function of \(f\). \(f(x)=4-x^{2}, \quad x \geq 0\)

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

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=\sqrt{x}, \quad g(x)=\frac{x}{x-1}, \quad h(x)=\sqrt[3]{x} $$

\(41-44=\) Find the domain and range of the function. $$ f(x)=-3 x^{2}+6 x+4 $$

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

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. $$ y=\frac{1}{5} x^{3}-1 $$

Find the inverse function of \(f\). \(f(x)=\sqrt{2 x-1}\)

Find the domain of the function. $$ f(x)=\frac{x^{4}}{x^{2}+x-6} $$

\(45-50\) Express the function in the form \(f \circ g\) $$ F(x)=(x-9)^{5} $$

\(45-46=\) 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). $$ f(x)=x^{2}+1.79 x-3.21 $$

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

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. $$ y=|x|-1 $$