Give you the same values so that the function maps, from one input to the same output. If you are in two of these intervals, the intervals should So it's very important that when you input - 5 in here, you know which 5 into the function, this thing would be filled in, and then the function wouldД«e defined both places and that's not cool for a function, it wouldn't be a function anymore. Important that this isn't a -5 is less than or equal to. Here, that at x equals -5, for it to be defined only one place. Over that interval, theДЇunction is equal to, the function is a constant 6. The next interval isДЇrom -5 is less than x, which is less than or equal to -1. If it was less than orД®qual, then the function would have been defined at This says, -9 is less than x, not less than or equal. It's a little confusing because the value of the function is actually also the value of the lower bound on this Over this interval? Well we see, the value That's this interval, and what is the value of the function I could write that as -9 is less than x, less than or equal to -5. X being greater than -9 and all the way up to and including -5. Is from, not including -9, and I have this open circle here.

So let me give myself some space for the three different intervals.

Then, let's see, our functionДЇ(x) is going to be equal to, there's three different intervals. Over here is the x-axis and this is the y=f(x) axis. Let's think about how we would write this using our function notation. In this interval for x, and then it jumps back downДЇor this interval for x. This graph, you can see that the function is constant over this interval, 4x. View them as a piecewise, or these types of function definitions they might be called a But what we're now going to explore is functions that areДefined piece by piece over different intervals By now we're used to seeing functions defined like h(y)=y^2 or f(x)= to the square root of x.

Piecewise function definition