And x starts off with -1 less than x, because you have an openĬircle right over here and that's good because X equals -1 is defined up here, all the way to x is 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 wouldīe 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, theįunction is equal to, the function is a constant 6. The next interval isįrom -5 is less than x, which is less than or equal to -1. If it was less than orĮqual, 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 functionį(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 downįor 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 areĭefined 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. If not, can anyone point me to a lesson where they are explained or at least mentioned earlier than the 'Domain of a function' lesson in Algebra I? I can't find them mentioned on this playlist, for example: - and definitely not anywhere earlier than the exercise I mentioned. Is this the first time piecewise functions are explained in the Khan Academy lessons? If so, I think some of the problems in the set I linked, or at least the Hint text for them, might be out of place. It's interesting (and kind of cool) that this video just came out as I've been looking for it. I have only been able to find it in the Algebra II lessons.

I have been looking and looking for Algebra I content that mentions piecewise functions, to make sure I learn it at the earliest point that I should have learned it. The Hint text says, 'f(x) is a piecewise function, so we need to examine where each piece is undefined.' (and goes on from there). One or more of the questions is all about the domain for a piecewise function. There is an exercise in the Algebra I content- 'Domain of a function' ( ). (old question kept for historical purposes) There were about 10 new modules added in the Algebra I 'Functions' section, I believe. These are a couple of the new modules added to address this: I believe those new modules added significant value to the lessons in that section of the KA content. Great job, Khan Academy! I am enjoying the new exercises, and I feel they really help fill in some small gaps that were there in the content. So this piece wise stuff may seem arcane or just a very special (infrequent) case, but it is not, it is a fixture in the mathematical landscape, so enjoy the view!Įdit: The Algebra I section has been expanded to include some modules that fill in these gaps nicely, and a few others. Each phase, launch, staging, orbit insertion, course correction and docking is a piece that has a very different characteristics of fuel consumption, and will require a different expression with different variables (air resistance, weight, gravity, burn rates etc.) at each stage in order to model it correctly. Hmmm, something more scientific? How about modeling the fuel usage of a space shuttle from launch to docking with the ISS. Or perhaps your local video store: rent a game, $5/per game, rent 2-3 games, $4/game, rent more than 5 games, $3/per game.Īsk your folks about tax brackets, another piece-wise function. In your day to day life, a piece wise function might be found at the local car wash: $5 for a compact, $7.50 for a midsize sedan, $10 for an SUV, $20 for a Hummer. Where ever input thresholds (or boundaries) require significant changes in output modeling, you will find piece-wise functions.

Desmos graphing piecewise functions