Chapter 1

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

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=\sqrt{x}, \quad g(x)=\sqrt{x}$$

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

Solve for \(y\) and use a graphing utility to graph each of the resulting equations in the same viewing window. (Adjust the viewing window so that the circle appears circular.) \((x-3)^{2}+(y-1)^{2}=25\)

Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$\left(\frac{3}{4}, \frac{3}{2}\right),\left(-\frac{4}{3}, \frac{7}{4}\right)$$

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=\frac{1}{x+8}-9$$.

Sketch the graph of the piecewise-defined function by hand. $$f(x)=\left\\{\begin{array}{ll} x+3, & x \leq 0 \\ 3, & 0 < x \leq 2 \\ 2 x-1, & x >2 \end{array}\right.$$

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=x^{2 / 3}, \quad g(x)=x^{6}$$

Find the domain of the function. $$h(t)=\frac{4}{t}$$

Determining Solution Points In Exercises 59 and 60 determine which point lies on the graph of the circle. (There may be more than one correct answer.) \((x-1)^{2}+(y-2)^{2}=25\) (a) (1,3) (b) (-2,6) (c) (5,-1) (d) \((0,2+2 \sqrt{6})\)

Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$(1,0.6),(-2,-0.6)$$

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=\frac{1}{x-7}+4$$.

Sketch the graph of the piecewise-defined function by hand. $$g(x)=\left\\{\begin{array}{ll} x+5, & x \leq-3 \\ 5, & -3 < x <1 \\ 5 x-4, & x \geq 1 \end{array}\right.$$

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=|x|, \quad g(x)=-x^{2}+1$$

Find the domain of the function. $$s(y)=\frac{3 y}{y+5}$$

Determining Solution Points In Exercises 59 and 60 determine which point lies on the graph of the circle. (There may be more than one correct answer.) \((x+2)^{2}+(y-3)^{2}=25\) (a) (-2,3) (b) (0,0) (c) (1,-1) (d) \((-1,3-2 \sqrt{6})\)

Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$(-8,0.6),(2,-2.4)$$

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=-2|x-1|-4$$.

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

\((a)\) find \((f \circ g)(x)\) and \((g \circ f)(x),\) \((b)\) determine algebraically whether \((f \circ g)(x)=(g \circ f)(x),\) and \((c)\) use a graphing utility to complete a table of values for the two compositions to confirm your answer to part \((b).\) $$f(x)=5 x+4, \quad g(x)=\frac{1}{5}(x-4)$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=\frac{3 x+4}{5}$$

Find the domain of the function. $$f(x)=\sqrt[3]{x-4}$$

A hospital purchases a new magnetic resonance imaging (MRI) machine for \(\$ 500,000. The depreciated value (reduced value) \)y\( after \)t\( years is \)y=500,000-47,000 t,\( for \)0 \leq t \leq 9\( (a) Use the constraints of the model and a graphing utility to graph the equation using an appropriate viewing window. (b) Use the value feature of the graphing utility to determine the value of \)y\( when \)t=5.8 .\( Verify your answer algebraically. (c) Use the zoom and trace features of the graphing utility to determine the value of \)t\( when \)y=156,900 .$ Verify your answer algebraically.

Use a graphing utility to graph the equation using each viewing window. Describe the differences in the graphs. $$y=0.25 x-2$$ $$\begin{array}{|l|l|l|} \hline \mathrm{Xmin}=-1 & \mathrm{Xmin}=-5 & \mathrm{Xmin}=-5 \\ \mathrm{Xmax}=9 & \mathrm{Xmax}=10 & \mathrm{Xmax}=10 \\ \mathrm{Xscl}=1 & \mathrm{Xscl}=1 & \mathrm{Xscl}=1 \\ \mathrm{Ymin}=-5 & \mathrm{Ymin}=-3 & \mathrm{Ymin}=-5 \\ \mathrm{Ymax}=4 & \mathrm{Ymax}=4 & \mathrm{Ymax}=5 \\ \mathrm{Yscl}=1 & \mathrm{Yscl}=1 & \mathrm{Yscl}=1 \\ \hline \end{array}$$

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=\frac{1}{2}|x-2|-3$$.

Sketch the graph of the piecewise-defined function by hand. $$h(x)=\left\\{\begin{array}{ll} 3+x, & x<0 \\ x^{2}+1, & x \geq 0 \end{array}\right.$$

\((a)\) find \((f \circ g)(x)\) and \((g \circ f)(x),\) \((b)\) determine algebraically whether \((f \circ g)(x)=(g \circ f)(x),\) and \((c)\) use a graphing utility to complete a table of values for the two compositions to confirm your answer to part \((b).\) $$f(x)=\frac{1}{4}(x-1), \quad g(x)=4 x+1$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=3 x+5$$

Find the domain of the function. $$f(x)=\sqrt[4]{x^{2}+3 x}$$

You buy a personal watercraft for \(\$ 8250 . The depreciated value \)y\( after \)t\( years is y=8250-689 t\) for \(0 \leq t \leq 10 (a) Use the constraints of the model and a graphing utility to graph the equation using an appropriate viewing window. (b) Use the zoom and trace features of the graphing utility to determine the value of \)t\( when \)y=5545.25 .\( Verify your answer algebraically. (c) Use the value feature of the graphing utility to determine the value of \)y\( when \)t=5.5 .$ Verify your answer algebraically.

Use a graphing utility to graph the equation using each viewing window. Describe the differences in the graphs. $$y=-8 x+5$$ $$\begin{array}{|l|l|l|} \hline \mathrm{Xmin}=-5 & \mathrm{Xmin}=-5 & \mathrm{Xmin}=-5 \\ \mathrm{Xmax}=5 & \mathrm{Xmax}=10 & \mathrm{Xmax}=13 \\ \mathrm{Xscl}=1 & \mathrm{Xscl}=1 & \mathrm{Xscl}=1 \\ \mathrm{Ymin}=-10 & \mathrm{Ymin}=-80 & \mathrm{Ymin}=-2 \\ \mathrm{Ymax}=10 & \mathrm{Ymax}=80 & \mathrm{Ymax}=10 \\ \mathrm{Yscl}=1 & \mathrm{Yscl}=20 & \mathrm{Yscl}=1 \\ \hline \end{array}$$

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=5$$

\((a)\) find \((f \circ g)(x)\) and \((g \circ f)(x),\) \((b)\) determine algebraically whether \((f \circ g)(x)=(g \circ f)(x),\) and \((c)\) use a graphing utility to complete a table of values for the two compositions to confirm your answer to part \((b).\) $$f(x)=\sqrt{x+6}, \quad g(x)=x^{2}-5$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=\frac{1}{x^{2}}$$

Find the domain of the function. $$g(x)=\frac{1}{x}-\frac{3}{x+2}$$

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l} L_{1}:(0,-1),(5,9) \\ L_{2}:(0,3),(4,1) \end{array}$$

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=-3 \sqrt{x+1}-6$$.

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=-9$$

\((a)\) find \((f \circ g)(x)\) and \((g \circ f)(x),\) \((b)\) determine algebraically whether \((f \circ g)(x)=(g \circ f)(x),\) and \((c)\) use a graphing utility to complete a table of values for the two compositions to confirm your answer to part \((b).\) $$f(x)=x^{3}-4, \quad g(x)=\sqrt[3]{x+10}$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$h(x)=\frac{4}{x^{2}}$$

Find the domain of the function. $$h(x)=\frac{10}{x^{2}-2 x}$$

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l} L_{1}:(-2,-1),(1,5) \\ L_{2}:(1,3),(5,-5) \end{array}$$

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=3 x-2$$

\((a)\) find \((f \circ g)(x)\) and \((g \circ f)(x),\) \((b)\) determine algebraically whether \((f \circ g)(x)=(g \circ f)(x),\) and \((c)\) use a graphing utility to complete a table of values for the two compositions to confirm your answer to part \((b).\) $$f(x)=|x|, \quad g(x)=2 x^{3}$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=(x+3)^{2}, \quad x \geq-3$$

Determine whether the statement is true or false. Justify your answer. A parabola can have only one \(x\) -intercept.

Find the domain of the function. $$g(y)=\frac{y+2}{\sqrt{y-10}}$$

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l} L_{1}:(3,6),(-6,0) \\ L_{2}:(0,-1),\left(5, \frac{7}{3}\right) \end{array}$$

The depreciation \(D\) (in millions of dollars) of the WD-40 Company assets from 2009 through 2013 can be approximated by the function $$D(t)=1.9 \sqrt{t+3.7}$$,where \(t=0\) represents 2009.(a) Describe the transformation of the parent function \(f(t)=\sqrt{t}\). (b) Use a graphing utility to graph the model over the interval \(0 \leq t \leq 4\). (c) According to the model, in what year will the depreciation of WD-40 assets be approximately 6 million dollars? (d) Rewrite the function so that \(t=0\) represents 2011 . Explain how you got your answer.

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=4-5 x$$