Compressing functions | Mathematics III | High School Math | Khan Academy

• [Voiceover] G of x is a transformation of f of x. The graph here shows this is y is equal to f of x, the solid blue line. This is y is equal to g of x as a dashed red line. And they ask us, "What is g of x in terms of f of x?" And like always, pause the video and see if you can give a go at it, and then we're going to do it together.

All right, so when you immediately look at it, it looks like g of x is kind of a thinned-up version of f of x. It seems like if you were to compress it towards the center, that's what g of x looks like. But let's put a little bit more meat on that bone and see if we can identify corresponding points.

So, for example, if we were to look at f of negative six... That's, so f of negative six. That seems like it corresponds or it gives us the same value as f of negative six. So we want to find the corresponding points. We hit this minimum point; we're coming back up. Hit the minimum point; we're coming back up. It seems like the corresponding point right over there is g of negative three. So let's write that down.

Let's see, it looks like f of negative six is equal to g of negative three. These are corresponding points. If you apply the transformation at the point f equals, at the point negative six comma f of negative six, you get to the point negative three, g of negative three right over there. Let's do a couple more.

If you look at f of two, it looks like it corresponds to g of one. F of two corresponds to g of one. So let's write that down. F of two looks like it corresponds to g of one. And once again, I'm looking at where the functions hit the same value, and also optically, I'm just looking at, well, it looks like it's the same part of the function if we assumed g of x is a squeezed version of f of x.

And so, in general, it looks like for a given x, we could say f of x is going to be equal to g of... well, whatever you have it here, it seems like we have half the value over here. So g of x over two. Or if you wanted to think of it the other way, if you want to say g of x is going to be f of... well, whatever we have here, it's f of twice that. So f of two x.

And we see that that is one of the choices: g of x is equal to f of two x. Whatever the x that you input into g of x, you get that same value out of the function when you input two times that into f of x. These seem to validate that. It looks optically like that, and we shrunk it down.

One way to think about it when you take, when you multiply the input into a function by a number larger than one, it's going to compress. It's going to make things happen faster. The input to the function is going to increase or become negative faster, so it thins it up.

And if that doesn't make intuitive sense, you can also just try some of these values. And I encourage you to try more. Find the corresponding points where the f's and the g's seem to match up. And you'll see over and over again that to get the same value, you have to put two times as much into f as you have to put into g. Hopefully, that helps.