Problem 1
Answers:
It costs 1.72 dollars to send a letter that is 1.5 ounces
It costs 1.72 dollars  to send a letter that is 2 ounces
In short, both answers are  1.72 dollars
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Explanation:
The piecewise function may seem really ugly and complicated, and I can understand why many don't like them, but they aren't too bad once you get used to it.
A piecewise function is simply a collection of many functions glued together. Instead of listing four different functions, your teacher has combined them all into one super function of sorts.
Here are the rules for this function
If
, then F(w) = 1.15If
, then F(w) = 1.72If
, then F(w) = 2.29If
, then F(w) = 2.86
As you can see, the definition of F(w) will change depending on what w is.
If the weight is w = 1.5 ounces, then we're on the interval since 1.5 is between 1 and 2. So we go for the second definition shown above and we conclude that it will cost $1.72 to send this 1.5 ounce letter.
If you wanted, you could rewrite those four bulleted points into this equivalent form
If
, then the cost is $1.15 to send the letterIf
, then the cost is $1.72 to send the letterIf
, then the cost is $2.29 to send the letterIf
, then the cost is $2.86 to send the letter
which is a less mathematical way of stating it. It might also help to make it into a chart, which would help customers better who may not want to deal with math.
A letter that's 2 ounces will cost $1.72 as well since w = 2 makes true.
In short, both the 1.5 ounce and 2 ounce letters cost $1.72 each.
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Problem 2
The graph is shown below (attached image)
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Explanation:
We have what is called a step function, or a staircase function. Either name will work. The graph consists of the horizontal portion of the stairs, but the vertical parts of the stairs are not drawn (or else we wouldn't have a function due to the vertical line test).
Note the placement of the open and closed endpoints. An open endpoint excludes the value in question, while a closed endpoint includes the value.
For instance, we have an open endpoint at (1, 1.72) while there's a closed endpoint at (1, 1.15). Each closed endpoint is directly connected to the "or equal to" portion of the inequality sign, and each open endpoint does not have the "or equal to" portion.
As the graph shows, we basically have the a certain flat cost for the four different regions. This corresponds to the costs given in the piecewise function. Each height is drawn from the list {1.15, 1.72, 2.29, 2.86}
The point labels of A,B,C,D,E,F,G,H are optional. I only put them in there to help identify which endpoints are open or closed. Points on the left side of each segment, or horizontal stair component, are open endpoints. Points on the right side are closed endpoints.
