Chapter 3

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=-f(x)\)

For the following exercises, write the domain for the piecewise function in interval notation. \(f(x)=\left\\{\begin{array}{l}x^{2}-2 \text { if } x<1 \\ -x^{2}+2 \text { if } x>1\end{array}\right.\)

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f(-x)\)

Graph \(y=\frac{1}{x^{2}}\) on the viewing window [-0.5,-0.1] and \([0.1,0.5] .\) Determine the corresponding range for the viewing window. Show the graphs.

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=6 f(x)\)

Graph \(y=\frac{1}{x}\) on the viewing window [-0.5,-0.1] and \([0.1, \quad 0.5]\). Determine the corresponding range for the viewing window. Show the graphs.

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f(5 x)\)

Suppose the range of a function \(f\) is \([-5, \quad 8]\). What is the range of \(|f(x)| ?\)

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f(2 x)\)

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(f(g(8))\)

Create a function in which the range is all nonnegative real numbers.

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f\left(\frac{1}{3} x\right)\)

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(f(g(5))\)

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f\left(\frac{1}{5} x\right)\)

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(g(f(5))\)

The height \(h\) of a projectile is a function of the time \(t\) it is in the air. The height in feet for \(t\) seconds is given by the function \(h(t)=-16 t^{2}+96 t .\) What is the domain of the function? What does the domain mean in the context of the problem?

For the following exercises, determine whether the relation represents a function. \(\\{(-1,-1),(-2,-2),(-3,-3)\\}\)

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=3 f(-x)\)

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(g(f(3))\)

The cost in dollars of making \(x\) items is given by the function \(C(x)=10 x+500\) (a) The fixed cost is determined when zero items are produced. Find the fixed cost for this item. (b) What is the cost of making 25 items? (c) Suppose the maximum cost allowed is \(\$ 1500\). What are the domain and range of the cost function, \(C(x) ?\)

For the following exercises, determine whether the relation represents a function. \(\\{(3,4),(4,5),(5,6)\\}\)

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=-f(3 x)\)

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(f(f(4))\)

For the following exercises, determine whether the relation represents a function. \(\\{(2,5),(7,11),(15,8),(7,9)\\}\)

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=|x|\) is reflected over the \(y\) -axis and horizontally compressed by a factor of \(\frac{1}{4}\).

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(f(f(1))\)

For the following exercises, determine if the relation represented in table form represents \(y\) as a function of \(x\). $$ \begin{array}{|c|c|c|c|} \hline x & 5 & 10 & 15 \\ \hline y & 3 & 8 & 14 \\ \hline \end{array} $$

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=\sqrt{x}\) is reflected over the \(x\) -axis and horizontally stretched by a factor of 2 .

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(g(g(2))\)

For the following exercises, determine if the relation represented in table form represents \(y\) as a function of \(x\). $$ \begin{array}{|l|l|l|l|} \hline x & 5 & 10 & 15 \\ \hline y & 3 & 8 & 8 \\ \hline \end{array} $$

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=\frac{1}{x^{2}}\) is vertically compressed by a factor of \(\frac{1}{3}\), then shifted to the left 2 units and down 3 units.

For the following exercises, use the function values for \(f\) and \(g\) shown in \(\underline{\text { Table } 3}\) to evaluate each expression. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(g(g(6))\)

For the following exercises, determine if the relation represented in table form represents \(y\) as a function of \(x\). $$ \begin{array}{|c|c|c|c|} \hline x & 5 & 10 & 10 \\ \hline y & 3 & 8 & 14 \\ \hline \end{array} $$

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=\frac{1}{x}\) is vertically stretched by a factor of 8 , then shifted to the right 4 units and up 2 units.

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline-3 & 11 & -8 \\ \hline-2 & 9 & -3 \\ \hline-1 & 7 & 0 \\ \hline 0 & 5 & 1 \\ \hline 1 & 3 & 0 \\ \hline 2 & 1 & -3 \\ \hline 3 & -1 & -8 \\ \hline \end{array} $$ \((f \circ g)(1)\)

For the following exercises, use the function \(f\) represented in the table below. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\ \hline \end{array} $$ Evaluate \(f(3)\).

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=x^{2}\) is vertically compressed by a factor of \(\frac{1}{2},\) then shifted to the right 5 units and up 1 unit.

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline-3 & 11 & -8 \\ \hline-2 & 9 & -3 \\ \hline-1 & 7 & 0 \\ \hline 0 & 5 & 1 \\ \hline 1 & 3 & 0 \\ \hline 2 & 1 & -3 \\ \hline 3 & -1 & -8 \\ \hline \end{array} $$ \((f \circ g)(2)\)

For the following exercises, use the function \(f\) represented in the table below. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\ \hline \end{array} $$ Solve \(f(x)=1\).

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=4-2 x\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(g(x)=4(x+1)^{2}-5\)

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=8-3 x\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(g(x)=5(x+3)^{2}-2\)

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline-3 & 11 & -8 \\ \hline-2 & 9 & -3 \\ \hline-1 & 7 & 0 \\ \hline 0 & 5 & 1 \\ \hline 1 & 3 & 0 \\ \hline 2 & 1 & -3 \\ \hline 3 & -1 & -8 \\ \hline \end{array} $$ \((g \circ g)(1)\)

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=8 x^{2}-7 x+3\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(h(x)=-2|x-4|+3\)

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline-3 & 11 & -8 \\ \hline-2 & 9 & -3 \\ \hline-1 & 7 & 0 \\ \hline 0 & 5 & 1 \\ \hline 1 & 3 & 0 \\ \hline 2 & 1 & -3 \\ \hline 3 & -1 & -8 \\ \hline \end{array} $$ \((f \circ f)(3)\)

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=3+\sqrt{x+3}\)

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(k(x)=-3 \sqrt{x}-1\)

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). \(f(x)=4 x+8, \quad g(x)=7-x^{2}\)