Chapter 2

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & -1 \\\\-2 & 0 \\\\-1 & 1 \\\0 & 2 \\\1 & 3 \\\2 & 4 \\\3 & 5\end{array}$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{x-3}{2}, g(x)=2 x+3$$

The table lists the federal minimum wage rates for the years \(1981-2017\). Sketch a graph of the data as a piece wise-defined function. (Assume that wages take effect on January 1 of the first year of the interval.) $$\begin{array}{|l|l|} \hline \text { Year(s) } & \text { Wage } \\ 1981-89 & \$ 3.35 \\ 1990 & \$ 3.80 \\ 1991-95 & \$ 4.25 \\ 1996 & \$ 4.75 \\ 1997-2006 & \$ 5.15 \\ 2007 & \$ 5.85 \\ 2008-2009 & \$ 6.55 \\ 2010-2017 & \$ 7.25 \\ \hline \end{array}$$

Solve each group of equations and inequalities analytically. (a) \(|\pi x+8|=-4\) (b) \(|\pi x+8|<-4\) (c) \(|\pi x+8|>-4\)

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{4}-7 x^{2}+6$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=5, g(x)=x$$

Suppose that the charges for an international cellular phone call are \(\$ 0.50\) for the first minute and \(\$ 0.25\) for each additional minute. Assume that a fraction of a minute is rounded up. (a) Determine the cost of a phone call lasting 3.5 minutes. (b) Find a formula for a function \(f\) that computes the cost of a telephone call \(x\) minutes long, where \(0< x \leq 5\) (Hint: Express \(f\) as a piece wise- defined function.)

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=-2 x^{6}-8 x^{2}$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x}, g(x)=1-x$$

Lumber that is used to frame walls of houses is frequently sold in lengths that are multiples of 2 feet. If the length of a board is not exactly a multiple of 2 feet, there is often no charge for the additional length. For example, if a board measures at least 8 feet, but less than 10 feet, then the consumer is charged for only 8 feet. (a) Suppose that the cost of lumber is \(\$ 0.80\) every 2 feet. Find a formula for a function \(f\) that computes the cost of a board \(x\) feet long for \(6 \leq x \leq 18\). (b) Use a graphing calculator to graph \(f\). (c) Determine the costs of boards with lengths of 8.5 feet and 15.2 feet.

Solve each equation or inequality. $$|2 x+4|+2=10$$

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

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x-2}, g(x)=\sqrt{x}$$

An express-mail company charges \(\$ 25\) for a package weighing up to 2 pounds. For each additional pound or fraction of a pound, there is an additional charge of \(\$ 3 .\) Let \(D(x)\) represent the cost to send a package weighing \(x\) pounds. Graph \(y=D(x)\) for \(x\) in the interval \((0,6]\).

Solve each equation or inequality. $$3|4-3 x|-4=8$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=-x^{5}+2 x^{3}-3 x$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x+1}, g(x)=3-6 x$$

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 mph.

Solve each equation or inequality. $$5|x+3|-2=18$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{6}-4 x^{4}+5$$

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

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. $$\frac{1}{2}\left|-2 x+\frac{1}{2}\right|=\frac{3}{4}$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=8$$

If \(f(x)=k, k \neq 0\) is a constant function and \(g(x)=m x+b, m \neq 0\) is a linear function, then determine the range of the composition \((f \circ g)(x)\)

From 1990 to 2007 , the number of people newly infected with HIV in Sub Saharan Africa increased from 1.3 million to 2.7 million. From 2007 to \(2016,\) the number fell from 2.7 million to 1.5 million. (a) Use the data points \((1990,1.3),(2007,2.7),\) and \((2016,1.5)\) to write equations for the two line segments that describe these data in the intervals \([1990,2007]\) and \((2007,2016]\). (b) Give a piece wise-defined function \(f\) that describes the graph.

Solve each equation or inequality. $$|3(x-5)+2|+3=9$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=3 x^{5}-x^{3}+7 x$$

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 in Exercises \(65-68\). $$f(x)=4 x+2, g(x)=\frac{1}{4}(x-2)$$

Solve each equation or inequality. $$4.2|0.5-x|+1=3.1$$

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

Solve each equation or inequality. $$|3 x-1|<8$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=|5 x|$$

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 in Exercises \(65-68\). $$f(x)=\sqrt[3]{5 x+4}, g(x)=\frac{1}{5} x^{3}-\frac{4}{5}$$

Solve each equation or inequality. $$|15-x|<7$$

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) \(y=-f(-x)\)

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

Solve each equation or inequality. $$|7-4 x| \leq 11$$

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) \(y=-3 f(x)\)

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

In a square viewing window, graph \(Y_{1}=\sqrt[3]{\mathrm{X}-6}\) and \(\mathrm{Y}_{2}=\mathrm{X}^{3}+6,\) 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.

Solve each equation or inequality. $$|2 x-3|>1$$

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

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

Solve each equation or inequality. $$|4-3 x|>1$$

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 \(y=(x-h)^{2}+k ?\) (Graph cannot copy)

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

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{x^{3}+3 x}{x}$$

For each function find (a) \(f(x+h)\) and (b) \(f(x)+f(h)\) $$f(x)=x^{2}-4$$

Solve each equation or inequality. $$|-3 x+8| \geq 3$$