Example 5.3.A

Given the function

[latex]f(x)=\begin{cases}7x+3\text{ if }x<0\\7x+6\text{ if }x\ge{0}\end{cases}[/latex],

evaluate:

1. [latex]f (-1)[/latex]
2. [latex]f (0)[/latex]
3. [latex]f (2)[/latex]

Answer: 1.[latex]f(x)[/latex] is defined as [latex]7x+3[/latex] for [latex]x=-1\text{ becuase }-1<0[/latex].   Evaluate: [latex]f(-1)=7(-1)+3=-7+3=-4[/latex] 2. [latex]f(x)[/latex] is defined as [latex]7x+6[/latex] for [latex]x=0\text{ becuase }0\ge{0}[/latex]. Evaluate: [latex]f(0)=7(0)+6=0+6=6[/latex] 3. [latex]f(x)[/latex] is defined as [latex]7x+6[/latex] for [latex]x=2\text{ becuase }2\ge{0}[/latex]. Evaluate: [latex]f(2)=7(2)+6=14+6=20[/latex]

Example 5.3.B

A cell phone company uses the function below to determine the cost, [latex]C[/latex], in dollars for [latex]g[/latex] gigabytes of data transfer.

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

Answer:

To find the cost of using 1.5 gigabytes of data, C(1.5), we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

C(1.5) = $25

To find the cost of using 4 gigabytes of data, C(4), we see that our input of 4 is greater than 2, so we use the second formula.

Example 5.3.C

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, [latex]n[/latex], to the cost, [latex]C[/latex].

Answer: Two different formulas will be needed. For n-values under 10, C=5n. For values of n that are 10 or greater, C=50. C(n)=[latex]\begin{cases}{5n}\text{ if }{0}<{n}<{10}\\ 50\text{ if }{n}\ge 10\end{cases}[/latex]

Example 5.3.D

Sketch a graph of the function. Given the piecewise definition [latex]f(x)=\begin{cases}−x − 3\text{ if }x < −3\\ x + 3\text{ if } x \ge −3\end{cases}[/latex] Draw the graph of f. State the domain and range of the function.

Answer: First, graph the line [latex]f(x) = −x−3[/latex], erasing the part where x is greater than -3. Place an open circle at (-3,0). Now place the line [latex]f(x) = x+3[/latex] on the graph, starting at the point (-3,0). Note that for this portion of the graph, the point (-3,0) is included, so you can remove the open circle. The two graphs meet at the point (-3,0) The domain of this function is all real numbers because (-3,0)is not included as the endpoint of [latex]f(x) = −x−3[/latex], but it is included as the endpoint for [latex]f(x) = x+3[/latex]. The range of this function starts at [latex]f(x)=0[/latex] and includes 0,  and goes to infinity, so we would write this as [latex]x\ge0[/latex]

Example 5.3.E

An on-line comic book retailer charges shipping costs according to the following formula [latex-display]S(n)=\begin{cases}1.5n+2.5\text{ if }1\le{n}\le14\\0\text{ if }n\ge15\end{cases}[/latex-display] Draw a graph of the cost function.

Answer: First, draw the line [latex]S(n)=1.5n+2.5[/latex].  We can use transformations: this is a vertical stretch of the identity by a factor of 1.5, and a vertical shift by 2.5.

Now we can eliminate the portions of the graph that are not in the domain based on [latex]1\le{n}\le14[/latex] Last, add the constant function S(n)=0 for inputs greater than or equal to 15. Place closed dots on the ends of the graph to indicate the inclusion of the end points.