Chapter 2

Sketch a graph showing the distance a person is from home after \(x\) hours if he or she drives on a straight road at 40 mph to a park 20 miles away, remains at the park for 2 hours, and then returns home at a speed of \(20 \mathrm{mph}\).

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=x^{3}-4 x$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=x^{2}, \quad g(x)=\sqrt{1-x}$$

Sketch a graph that depicts the amount of water in a 100 -gallon tank. The tank is initially empty and then filled at a rate of 5 gallons per minute. Immediately after it is full, a pump is used to empty the tank at 2 gallons per minute.

Solve each equation or inequality. $$\left|6-\frac{1}{3} x\right|>0$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=x+2, \quad g(x)=x^{4}+x^{2}-3 x-4$$

Sub-Saharan HIV Infection Rates From 1990 to 2007 the number of people newly infected with HIV in SubSaharan Africa increased from 1.3 million to 2.7 million. From 2007 to \(2012,\) the number fell from 2.7 million to 1.75 million. A. Use the data points \((1990,1.3),(2007,2.7),\) and \((2012,1.75)\) to write equations for the two line segments describing these data in the closed intervals \([1990,2007]\) and \([2007,2012]\) B. Give a piecewise-defined function \(f\) that describes the graph.

Solve each equation or inequality. $$|8 x-4|<0$$

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=\sqrt{x^{2}+1}$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=\frac{1}{x+1}, \quad g(x)=5 x$$

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=\frac{1}{2 x}$$

Concept Check If \((r, 0)\) is an \(x\) -intercept of the graph of \(y=f(x),\) what statement can be made about an \(x\) -intercept of the graph of each function? (Hint: Make a sketch.) (a) \(y=-f(x)\) (b) \(y=f(-x)\) (c) \(y=-f(-x)\)

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=x+4, \quad g(x)=\sqrt{4-x^{2}}$$

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=4 x-\frac{1}{x}$$

Concept Check If \((0, b)\) is the \(y\) -intercept of the graph of \(y=f(x),\) what statement can be made about the \(y\) -intercept of the graph of each function? (Hint: Make a sketch.) (a) \(y=-f(x)\) (b) \(y=f(-x)\) (c) \(y=5 f(x)\) (d) \(y=-3 f(x)\)

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=2 x+1, \quad g(x)=4 x^{3}-5 x^{2}$$

Solve each equation or inequality. $$|7 x-5| \geq-5$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=-x^{3}+2 x$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$f(x-2)$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=\frac{x-3}{2}, \quad g(x)=2 x+3$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{5}-2 x^{3}$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$5 f(x+1)$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=5, \quad g(x)=x$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|3 x+1|=|2 x-7|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=0.5 x^{4}-2 x^{2}+1$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$-f(x)$$

If \(f(x)\) defines a constant function over \((-\infty, \infty),\) how many elements are in the range of \((f \circ f)(x) ?\)

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|x-4|=|7 x+12|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=0.75 x^{2}+|x|+1$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$f(x-3)+1$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=4 x+2, g(x)=\frac{1}{4}(x-2)$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|-2 x+5|=|x+3|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{3}-x+3$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$f(2 x)$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=-3 x, g(x)=-\frac{1}{3} x$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|-5 x+1|=|3 x-4|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{4}-5 x+2$$

Concept Check Suppose that the graph of \(y=x^{2}\) is translated in such a way that its domain is \((-\infty, \infty)\) and its range is \([38, \infty) .\) What values of \(h\) and \(k\) can be used if the new function is of the form \(y=(x-h)^{2}+k ?\)

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$2 f(x-1)$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=\sqrt[3]{5 x+4}, g(x)=\frac{1}{5} x^{3}-\frac{4}{5}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$\left|x-\frac{1}{2}\right|=\left|\frac{1}{2} x-2\right|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{6}-4 x^{3}$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$3 f\left(\frac{1}{4} x\right)$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=\sqrt[3]{x+1}, g(x)=x^{3}-1$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|x+3|=\left|\frac{1}{3} x+8\right|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{3}-3 x$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$-2 f(4 x)$$

Functions such as the pairs in Exercises \(69-72\) are called inverse functions, because the result of composition in both directions is the identity function. (Inverse functions will be discussed in detail in Section 5.1.) In a square viewing window, graph \(y_{1}=\sqrt[3]{x-6}\) and \(y_{2}=x^{3}+6,\) an example of a pair of inverse functions. Now graph \(y_{3}=x .\) Describe how the graph of \(y_{2}\) can be obtained from the graph of \(y_{1},\) using the graph \(y_{3}=x\) as a basis for your description.

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|4 x+1|=|4 x+6|$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=-6$$