Chapter 1

Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of \(g\) and \(h\) relative to the graph of \(f\).$$\begin{aligned}&f(x)=x^{3}-3 x^{2}\\\&g(x)=-\frac{1}{3} f(x)\\\&h(x)=f(-x)\end{aligned}$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$h(x)=\sqrt{16-x^{2}}$$

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{x+3}$$

Assume that the domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\} .\) Determine the set of ordered pairs representing the function \(f.\) $$f(x)=|x|+2$$

Sketch the graph of the function by hand. Then use a graphing utility to verify the graph. $$f(x)=[x+2]+1$$

Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of \(g\) and \(h\) relative to the graph of \(f\).$$\begin{aligned}&f(x)=x^{3}-3 x^{2}+2\\\ &g(x)=-f(x)\\\&h(x)=f(2 x)\end{aligned}$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$f(x)=-2 x \sqrt{16-x^{2}}$$

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{2 x}$$

Assume that the domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\} .\) Determine the set of ordered pairs representing the function \(f.\) $$f(x)=|x+1|$$

Sketch the graph of the function by hand. Then use a graphing utility to verify the graph. $$f(x)=2[[x]$$

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=|x-4|, \quad g(x)=3-x$$

Complete the table. $$h(t)=\frac{1}{2}|t+3|$$ $$\begin{array}{|l|l|l|l|l|l|} \hline t & -5 & -4 & -3 & -2 & -1 \\ \hline h(t) & & & & & \\ \hline \end{array}$$

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.) \(y=\sqrt{5-x}\) (a) \((3, y)\) (b) \((x, 3)\)

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). $$(5,-1),(-5,5)$$

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 \(f\).$$g(x)=(x-10)^{2}+5$$

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=\frac{2}{|x|} \quad g(x)=x-5$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$f(x)=-0.65$$

Complete the table. $$f(s)=\frac{|s-2|}{s-2}$$ $$\begin{array}{|l|l|l|l|l|l|} \hline s & 0 & 1 & \frac{3}{2} & \frac{5}{2} & 4 \\ \hline f(s) & & & & & \\ \hline \end{array}$$

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.) \(y=x^{2}(x-3)\) (a) \((-1, y)\) (b) \((x, 6)\)

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). $$(4,3),(-4,-4)$$

Use a graphing utility to graph the function. State the domain and range of the function. Describe the pattern of the graph. $$s(x)=2\left(\frac{1}{4} x-\left[\left[\frac{1}{4} x\right]\right)\right.$$

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=x+2, \quad g(x)=\frac{1}{x^{2}-4}$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$g(x)=(x+5)^{3}$$

Find all values of \(x\) such that \(f(x)=0.\) $$f(x)=15-3 x$$

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.) \(y=x^{5}-5 x\) (a) \((-0.5, y)\) (b) \((x,-2)\)

Use a graphing utility to graph the function. State the domain and range of the function. Describe the pattern of the graph. $$g(x)=2\left(\frac{1}{4} x-\left[\left[\frac{1}{4} x\right]\right)^{2}\right.$$

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=\frac{3}{x^{2}-1}, \quad g(x)=x+1$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$f(x)=x^{5}-7$$

Find all values of \(x\) such that \(f(x)=0.\) $$f(x)=5 x+1$$

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.) \(y=\left|x^{2}-6 x+5\right|\) (a) \((2, y)\) (b) \((x, 1.5)\)

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). $$(-1,6),(5,6)$$

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

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 \(f\).$$g(x)=\frac{1}{3}(x-2)^{3}$$.

(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 \(f \circ g=g \circ f.\) $$f(x)=\sqrt{x+4}, \quad g(x)=x^{2}$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$h(x)=|x|-|x-4|$$

Find all values of \(x\) such that \(f(x)=0.\) $$f(x)=\frac{9 x-4}{5}$$

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.) \(x^{2}+y^{2}=16\)

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). $$\left(2, \frac{1}{2}\right),\left(\frac{1}{2}, \frac{5}{4}\right)$$

Sketch the graph of the piecewise-defined function by hand. $$f(x)=\left\\{\begin{array}{ll} x+6, & x \leq-4 \\ 3 x-4, & x>-4 \end{array}\right.$$

(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 \(f \circ g=g \circ f.\) $$f(x)=\sqrt[3]{x+1}, \quad g(x)=x^{3}-1$$

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}$$

Find all values of \(x\) such that \(f(x)=0.\) $$f(x)=\frac{2 x-3}{7}$$

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.) \(x^{2}+y^{2}=36\)

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). $$(1,1),\left(6,-\frac{2}{3}\right)$$

Sketch the graph of the piecewise-defined function by hand. $$f(x)=\left\\{\begin{array}{ll} \sqrt{4+x}, & x<0 \\ \sqrt{4-x}, & x \geq 0 \end{array}\right.$$

(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 \(f \circ g=g \circ f.\) $$f(x)=\frac{1}{3} x-3, \quad g(x)=3 x+9$$

Find the domain of the function. $$f(x)=5 x^{2}+2 x-1$$

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.) \((x-1)^{2}+(y-2)^{2}=49\)

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). $$\left(-\frac{1}{10},-\frac{3}{5}\right),\left(\frac{9}{10},-\frac{9}{5}\right)$$

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 \(f\).$$g(x)=-(x+3)^{3}-10$$.