# Jeepneys and Step Functions

In the Philippines, jeepneys are the kings of the road. They are our major way of public transport and, of course, the cheapest.  As of this writing, the jeepney fare in the Philippines is about Php8.00 (about US\$0.19) for the first four kilometers and an additional 1.25 pesos (about US\$ 0.03)  for each succeeding kilometer.

The table below shows the jeepney fare matrix for the first 10 kilometers. In the matrix, d is the distance traveled in kilometers and F be the corresponding fare.

It is clear from the table that for each distance traveled, there is one and only one corresponding fare; therefore, fare is a function of the distance traveled by the jeepney. The graph of the values in the table is shown below. For distance greater than 0 up to 4 kilometers, the fare is Php8.00. Since there is no corresponding fare for d=0, we represent the point (0,F(0)), and the leftmost part of each horizontal segment with an empty circle. An empty circle indicates that the function does not have a value on that particular point; that is, F is undefined. On the other hand, a point exists for (4,F(4)), here F(4) = 8, but an empty circle is also present on the same x,  just “one step higher.” This means that for 4 kilometers, the fare is still Php8.00 and beyond that, it will already be Php9.25. As you can observe, this is also the same for the other intervals.

The function above is a step function because the value of the dependent variable F jumps or steps from one value to another. Step functions are a special type of piecewise function.

In the context above, what is the domain and range of F?

Suppose that a jeepney can travel in an infinitely long road, what will be the domain and range of F?

*1 US dollar equals Php43.00as of this writing.