We will be interpreting and graphing piecewise functions. Now you may think to yourself there are no real life applications for piecewise functions, but in fact if you own a cell phone or if you've ever shipped a package through UPS or through the US Postal Service, those rates are determined by piecewise functions. So that's pretty exciting stuff. Let's go ahead and jump into an example. In 2005, Sprint offered a cell phone calling plan in which a customer's monthly bill can be modeled by the graph below.
The amount of the bill is a function f of the number of minutes of phone use. So down here on the x-axis we have the number of minutes used per month. And on the y-axis we have the monthly bill in dollar.
Under this plan if a customer uses the phone for 360 minutes, What is his or her monthly bill? I'll go back to the larger graph on the previous screen. Locate 360 on the x-axis.
Go straight to the graph. The graph is defined here where y is equal to 40. So the bill would be $40. I would like to go back and take a look at another case before we move on to the next question.
What if for example you use 700 minutes? As you go up to the graph, notice how there's a closed point here and there's an open point here. Well where is the function defined? Remember the function is defined at the closed point, which would result in a y value of fifty-five dollars. The function is not defined at the open points, just to review.
The next question. If a monthly bill is $55, for how many minutes did the customer use the phone? Well we just kind of started analyzing that. If the person paid $55, they could have used...
Notice there's a horizontal interval where the bill is $55. In fact, if a customer used, let's say, 640 minutes. the bill would be $55.
620 minutes would still be a bill of $55. Now 600 is right kind of at the dividing line. If the customer used 600 minutes, now the bill would be $50.
But if they used 601 minutes, it would jump up to $55. In fact, if it was 600.001 minutes, it would jump to the next interval. So how do we express that? Let's go back to the question.
If we let t equal the number of minutes used, we know that it has to be less than or equal to 700. But t also has to have a lower bound, and it has to be greater than 600. Notice how this inequality symbol, it does not include the value of 600. So the interval from 600 to 700, including 700, not including 600. Let's take a look at the graph of piecewise functions. For example, graph the function defined as follows. g of x is equal to 2 for x greater than or equal to 1. g of x equal to negative x minus 2 for x less than 1. There are two pieces to this function so the end result may have two pieces.
Here's how we want to interpret this. g of x is equal to 2, and that's the horizontal line y equals 2, but only for x greater than or equal to 1. So x is greater than or equal to 1 starting at 1 and moving to the right. Now the second piece of this function is the line g of x.
is equal to negative x minus 2 only for x less than 1. So remember this line is a y intercept of negative 2, slope of negative 1. It would be this red line, but notice how we stopped it when x was equal to 1, and since it did not include 1, we made an open circle to show that, and then only the graph on the left side of x equals 1. Now, So these two graphs make up the function g of x. So we can't keep them separate, we have to combine them onto the same coordinate plane to look like this. Okay, hopefully that explains this piecewise function. Let's take a look at one more.
Now this one looks like it may have three pieces and in fact it does. And I'm going to show this one also on the graphing calculator. But let's kind of explain this interval by interval.
The first interval, we have the horizontal line f of x equals three when x is less than or equal to zero. Here's the horizontal line y equals 3 only when x is less than 0. The next piece, we want the function f of x equals 1 minus x squared. Remember that's a parabola.
We only want a small piece of it from 0 to 2, not including 0, but including 2. So notice how we have an open circle when x is 0 and we could put a closed point if we wanted to. right there where x is 2. I think I'll do that just to emphasize that. In the last piece we have the line f of x equals 2x minus 4, this black line. We want that black line when x is greater than 2, not including 2. So we have an open circle at x equals 2 and then greater than meaning going to the right.
We have three pieces. Now if we want to graph this on our graphing calculators we can. We have to enter it in three different times. Let's take a look at that. The way we do this is we go to our y equals screen.
Okay let's take a look at how we would graph this on the graphing calculator. It is kind of involved on the graphing calculator as well. It is by no means an easy process and the general procedure is this. Let's go ahead and hit y equals where you're on the screen that I'm on. First you type in your given rule or function for each piece and then you multiply it by in parentheses the interval for which you want to consider that graph.
So for example the first function was f of x equals 3 so I typed in the 3 and then I multiplied it by x is less than or equal to 0. Now the second piece of the function 1 minus x squared since I have a sum or difference of two terms, I do have to put that function rule in a set of parentheses. And I multiply it by, in this case, I have two intervals to multiply it by. You cannot type this in as a compound interval or compound inequality in the calculator. So I had to type in 0 less than x in parentheses and x greater than or equal to 2. I'll show you how to type in these inequalities in just a moment. And the third piece, the line 2x minus 4 in parentheses, and then multiply that by x greater than 2. Now, in order to find these inequality symbols, it's a pretty easy process.
Just hit second math, which brings up the test menu, and you pick those options that you need. Now, I'm not going to go through typing each one of these in due to time, but hopefully that will give you a good idea of how you can do this. Another drawback to the graphing calculator is when you do graph it, it does not include the open and closed points. You still have to determine that by