Chapter 3

Sketch the graph of the function by first making a table of values. $$ f(x)=\frac{x}{|x|} $$

If \(g(x)=x^{2}+4 x\) with \(x \geq-2,\) find \(g^{-1}(5)\)

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(z)=1-3 z^{2} ; \quad z=-2, z=0 $$

Evaluate the function at the indicated values. $$ \begin{array}{l}{f(x)=\frac{|x|}{x}} \\ {f(-2), f(-1), f(0), f(5), f\left(x^{2}\right), f\left(\frac{1}{x}\right)}\end{array} $$

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ g)(x)} & {\text { (b) }(g \circ f)(x)}\end{array} $$

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ f(x)=x^{2}+2 x-1 $$

Sketch the graph of the function by first making a table of values. $$ g(x)=\frac{2}{x^{2}} $$

(a) Sketch the graph of \(f(x)=\frac{1}{x}\) by plotting points. (b) Use the graph of \(f\) to sketch the graphs of the following functions. (i) \(y=-\frac{1}{x} \quad\) (ii) \(y=\frac{1}{x-1}\) (iii) \(y=\frac{2}{x+2} \quad\) (iv) \(y=1+\frac{1}{x-3}\)

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=x-6, \quad g(x)=x+6\)

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=x^{3}-4 x^{2} ; \quad x=0, x=10 $$

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

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ f)(x)} & {\text { (b) }(g \circ g)(x)}\end{array} $$

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ f(x)=x^{2}-8 x+8 $$

Sketch the graph of the function by first making a table of values. $$ g(x)=\frac{|x|}{x^{2}} $$

(a) Sketch the graph of \(g(x)=\sqrt[3]{x}\) by plotting points. (b) Use the graph of \(g\) to sketch the graphs of the following functions. \(\begin{array}{ll}{\text { (i) } y=\sqrt[3]{x-2}} & {\text { (ii) } y=\sqrt[3]{x+2}+2} \\ {\text { (iii) } y=1-\sqrt[3]{x}} & {\text { (iv) } y=2 \sqrt[3]{x}}\end{array}\)

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

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=x+x^{4} ; \quad x=-1, x=3 $$

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

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ f(x)=-x^{2}-3 x+3 $$

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=x^{2}, \quad g(x)=(x+2)^{2}\) (b) \(f(x)=x^{2}, \quad g(x)=x^{2}+2\)

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=2 x-5 ; \quad g(x)=\frac{x+5}{2}\)

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=3 x^{2} ; \quad x=2, x=2+h $$

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

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ f(x)=1-6 x-x^{2} $$

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=x^{3}, \quad g(x)=(x-4)^{3}\) (b) \(f(x)=x^{3}, \quad g(x)=x^{3}-4\)

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

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=4-x^{2} ; \quad x=1, x=1+h $$

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

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ g(x)=3 x^{2}-12 x+13 $$

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=\sqrt{x}, \quad g(x)=2 \sqrt{x}\) (b) \(f(x)=\sqrt{x}, \quad g(x)=\frac{1}{2} \sqrt{x-2}\)

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

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

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ g(x)=2 x^{2}+8 x+11 $$

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=|x|, \quad g(x)=3|x|+1\) (b) \(f(x)=|x|, \quad g(x)=-|x+1|\)

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

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ g(x)=\frac{2}{x+1} ; \quad x=0, x=h $$

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

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ h(x)=1-x-x^{2} $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=x-1 $$

\(27-32\) : 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 and shift 2 units to the right

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

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(t)=\frac{2}{t} ; \quad t=a, t=a+h $$

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

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ h(x)=3-4 x-4 x^{2} $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=2(x+1) $$

\(27-32\) : 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 and shift 4 units to the left

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

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(t)=\sqrt{t} ; \quad t=a, t=a+h $$

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

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=2 x+3, \quad g(x)=4 x-1 $$