- [Instructor] We're told this is the graph of function f, fair enough. Function g is defined as g of x is equal to f of two x. What is the graph of g? So pause this video, and try to figure that out on your own. All right, now let's work through this. And the way I will think about it, I'll set up a little table here. And I'll have an x column, and then I'll have a, well, actually just put g of x column. And of course, g of x is equal to f of two x. So when x is, and actually let me see, when x is equal to, I could pick a point like x equaling zero, so g of zero is going to be f of two times zero. So it's going to be f of two times zero, which is still f of zero, which is going to be equal to a little bit over four, so which is equal to f of zero. And so they're going to both have the same y-intercept, but interesting things are going to happen the further that we get from the y-axis or as our x increases in either direction away, or as our x gets bigger in either direction from zero. So let's think about what's going to happen at x equals two. So at x equals two, g of two is going to be equal to f of two times two, two times two, which is equal to f of four. And we know what f of four is. F of four is equal to zero. So g of two is equal to f of four, which is equal to zero. So notice, the corresponding point has kind of gotten compressed in or squeezed in or squished in, in the horizontal direction. And so what you see happening, at least on this side of the graph, is everything's happening a little bit faster. Whatever was happening at a certain x, it's now happening at half of that x. So this side of the graph is going to look something, try to draw it a little bit better than that, it's going to look something like this, like this. Everything's happening twice as fast. And what happens when you go in the negative direction? Well, think about what g of negative two is. G of negative two is equal to f of two times negative two, two times negative two, which is equal to f of negative four, which we see is also equal to zero. So g of negative two is zero. And you might be thinking, "Why did you pick two and negative two?" Well, the intuition is that things are going to be squeezed in. Things are happening twice as fast. So whatever was happening at x equals four is now going to happen at x equals two. Whatever is happening at x equals negative four is now going to happen at x equals negative two. And I saw that we were at very clear points at x equals negative four and x equals four on f, so I just took half of that to pick my x-values right over here. And then so what our graph is going to look like is something like this. It's going to look something like this. It's going to look like it's been squished in from the right and the left. Now let's do another example. So now they've not only given the graph of f, they've given an expression for it. What is the graph of g of x which is equal to this business? So pause this video, and try to figure that out. All right, the key is to figure out the relationship between f of x and g of x. And what we can see, the main difference is, is instead of an x here in f of x, we have an x over two. So everywhere there was an x, we've been replaced with an x over two. So another way of thinking about it is g of x is equal to f of not x but f of x over two. Or another way of thinking about it, g of x is equal to f of 1/2x. And then we can do a similar type of exercise. And they've given us some interesting points, the points two, the point, or the point x equals two, the point x equals four, and the point x equals six. So let's think about this. Last time, when it was g of x is equal to two x, things were happening twice as fast. Now things are going to happen half as fast. And so what I would do, let me just set up a little table here. The interesting x-values for me are the ones that if I take half of them, then I'm going to get one of these points. So actually let me write this, half, 1/2x, and then I can think about what g of x is equal to f of 1/2x is going to be. So I want my 1/2x to be, let's see, it could be two, four, and six, two, four, and six. And why did I pick those again? Well, it's very clear what values f takes on at those points. And so if 1/2x is two, then x is equal to four. If 1/2x is four, then x is equal to eight. If x is equal to 12, then 1/2x is six. And so then we could say, all right, g of four is equal to f of two, which is equal to zero. That's why I picked two, four, and six. It's very easy to evaluate f of two, f of four, and f of six. They gave us those points very clearly. So g of eight is going to be equal to g, is going to be equal to f of 1/2 of eight, or f of four, which is equal to negative four. And then g of 12 is equal to f of six, which is half of 12, which is equal to zero again. So then we could plot these points, and we get a general sense of the shape of the graph. So let's see, g of four is equal to zero, g of eight is equal to negative four right over there, and then g of 12 is equal to zero again. So everything has been stretched out. So there you go, it's been stretched out in at least, in the horizontal direction is one way to think about it, in the horizontal direction. And you can see that this point in f corresponds to this point in g. It's gotten twice as far from the origin because everything is growing half as fast. You input an x, you take a half of it, and then you input it into f. And then this point right over here corresponds to this point. Instead of happening at four, this vertex point, it's now happening at eight. And last but not least, this point right over here corresponds to this point. Instead of happening at six, it's happening at 12. Everything is getting stretched out. Let's do one more example. F of x is equal to all of this. We have to be careful, there's a cube root over here. And g is a horizontally scaled version of f. The functions are graphed where f is solid and g is dashed. What is the equation of g? So pause this video, and see if you can figure that out. All right, let's do this together, and it looks like they've given us some points that seem to correspond with each other. To go from f to g, it looks like these corresponding points have been squeezed in closer to the origin. And what we can see is, is that f of negative three, f of negative three seems to be equal to g of negative one. And f of six over here, f of six seems to be equal to g of two, g of two. Or another way to think about it, whatever x you input in g, it looks like that's going to be equivalent to three times that x inputted into f. So g of x is equal to f of three x. And so if you want to know the equation of g, we just evaluate f of three x. So f of three x is going to be equal to, and I could just actually put an equal sign like this, f of three x is going to be equal to negative three times the cube root of, instead of an x, I'll put a three x right over there, three x plus two, and then we have plus one. And that's it, that's what g of x is equal to. It's equal to f of three x, which is that. We substituted this x with a three x. And we are done.