Chapter 1

\((a)\) 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 \((b).\) $$f(x)=\frac{6}{3 x-5}, \quad g(x)=-x$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$q(x)=(x-5)^{2}, \quad x \leq 5$$

Determine whether the statement is true or false. Justify your answer. The graph of a linear equation can have either no \(x\) -intercepts or only one \(x\) -intercept.

Find the domain of the function. $$f(x)=\frac{\sqrt{x+6}}{6+x}$$

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l} L_{1}:(4,8),(-4,2) \\ L_{2}:(3,-5),\left(-1, \frac{1}{3}\right) \end{array}$$

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$h(x)=x^{2}+6$$

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

Determine whether the statement is true or false. Justify your answer. Writing Your employer offers you a choice of wage scales: a monthly salary of \(\$ 3000\) plus commission of \(7 \%\) of sales or a salary of \(\$ 3400\) plus a \(5 \%\) commission. Write a short paragraph discussing how you would choose your option. At what sales level would the options yield the same salary?

Use a graphing utility to graph the function. Find the domain and range of the function. $$f(x)=\sqrt{16-x^{2}}$$

Determine whether the statement is true or false. Justify your answer.The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=-x^{2}-8$$

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

Use a graphing utility to graph the function. Find the domain and range of the function. $$f(x)=\sqrt{x^{2}+1}$$

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=f(-x)$$.

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=\sqrt{1-x}$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=|x-2|, \quad x \leq 2$$

Perform the operation and write the result in standard form. \((9 x-4)+\left(2 x^{2}-x+15\right)\)

Use a graphing utility to graph the function. Find the domain and range of the function. $$g(x)=|2 x+3|$$

Write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. $$\left(-\frac{2}{3}, \frac{7}{8}\right), \quad 3 x+4 y=7$$

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=2 f(x)$$.

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$g(t)=\sqrt[3]{t-1}$$

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=\frac{x^{2}}{x^{2}+1}, \quad x \geq 0$$

Perform the operation and write the result in standard form. \(\left(3 x^{2}-5\right)\left(-x^{2}+1\right)\)

Use a graphing utility to graph the function. Find the domain and range of the function. $$g(x)=|3 x-5|$$

Write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. $$\left(\frac{2}{5},-1\right), \quad 3 x-2 y=6$$

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=f(x)+2$$.

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

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) (There are many correct answers.) $$h(x)=(2 x+1)^{2}$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=4 x-9$$

Write the area \(A\) of a circle as a function of its circumference \(C .\)

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=f(x-3)$$.

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

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) (There are many correct answers.) $$h(x)=(1-x)^{3}$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=3 x$$

Write the area \(A\) of an equilateral triangle as a function of the length \(s\) of its sides.

Write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. $$(-1.2,2.4), \quad 5 x+4 y=1$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{5}$$

An open box of maximum volume is to be made from a square piece of material, 24 centimeters on a side, by cutting equal squares from the corners and turning up the sides. (See figure.) (a) The table shows the volume \(V\) (in cubic centimeters) of the box for various heights \(x\) (in centimeters). Use the table to estimate the maximum volume. $$\begin{array}{|c|c|} \hline \text { Teather } & \text { Volumes } \\ \hline 1 & 484 \\ 2 & 800 \\ 3 & 972 \\ 4 & 1024 \\ 5 & 980 \\ 6 & 864 \\ \hline \end{array}$$ (b) Plot the points \((x, V)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(V\) as a function of \(x ?\) (c) If \(V\) is a function of \(x,\) write the function and determine its domain. (d) Use a graphing utility to plot the points from the table in part (a) with the function from part (c). How closely does the function represent the data? Explain.

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{3}+1$$

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) (There are many correct answers.) $$h(x)=\sqrt{9-x}$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{4}, \quad x \leq 0$$

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) (There are many correct answers.) $$h(x)=\frac{1}{x+2}$$

A rectangle is bounded by the \(x\) -axis and the semicircle \(y=\sqrt{36-x^{2}},\) as shown in the figure. Write the area \(A\) of the rectangle as a function of \(x\) and determine the domain of the function.

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{2}, \quad x \geq 0$$

You can use either of two methods to graph a function: plotting points, or translating a parent function as shown in this section. Which method do you prefer to use for each function? Explain. $$\text { (a) } f(x)=3 x^{2}-4 x+1$$ $$\text { (b) } f(x)=2(x-1)^{2}-6$$

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=\sqrt{4-x^{2}}, \quad 0 \leq x \leq 2$$

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\) (There are many correct answers.) $$h(x)=(x+4)^{2}+2(x+4)$$

A company produces a product for which the variable cost is \(\$ 68.75\) per unit and the fixed costs are \(\$ 248,000 .\) The product sells for \(\$ 99.99 .\) Let \(x\) be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Write the profit \(P\) as a function of the number of units sold. (Note: \(P=R-C.)\) (d) Use the model in part (c) to find \(P(20,000) .\) Interpret your result in the context of the situation. (e) Use the model in part (c) to find \(P(0) .\) Interpret your result in the context of the situation.

The graph of \(y=f(x)\) passes through the points \((0,1),(1,2),\) and \((2,3) .\) Find the corresponding points on the graph of \(y=f(x+2)-1\).

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=\sqrt{16-x^{2}}, \quad-4 \leq x \leq 0$$