- [Instructor] What we're going to do in this video is do some practice examples of exercises on Khan Academy that deal with reflections of functions. So this first one says this is the graph of function f. Fair enough. Function g is defined as g of x is equal to f of negative x. Also fair enough. What is the graph of g? And on Khan Academy, it's multiple choice, but I thought for the sake of this video, it'd be fun to think about what g would look like without having any choices, just sketching it out. So pause this video and try to think about it, at least in your head. All right, now let's work through this together. So we've already gone over that g of x is equal to f of negative x. So whatever the value of f is at a certain value, we would expect g to take on that value at the negative of that. So for example, we can see that f of four is equal to two, so we would expect g of negative four to be equal to two because, once again, g of negative four, we could write it over here. G of negative four is going to be equal to f of the negative of negative four, which is equal to f of four. And so we could keep going with that. What would g of negative two be? Well, that would be the same thing as f of two, which is zero, so it would be right over there. What would g of zero be? Well, that would be the same thing as f of zero 'cause a negative zero is zero. And f of zero is right over there. Looks like negative two. And so you can already see where this is going. And we've already talked about it in previous videos that if you replace your x with a negative x, you're essentially reflecting over the y-axis. So g is going to look something like this. It is going to look something like this. Once again, g of negative six would be the same thing as f of six. And so that would be the graph of g. And if you're doing this on Khan Academy, you'd pick the choice that looks like this, that would give a reflection over the y-axis. Let's do another example. So here once again, this is the graph of the function f. And then they say what is the graph of g? And so pause this video and at least try to sketch it out in your mind what g should look like. All right, so in this situation, they didn't replace the x with negative x in f of x. Instead, g of x is equal to the negative of all of f of x. In fact, we could rewrite g of x like this. We could say that g of x is equal to, notice, all of this right over here, that was our definition of f of x. So g of x is equal to the negative of f of x. So instead of it being f of negative x, it's equal to the negative of f of x. So one way to think about it is we can see that f of zero is two, but g of zero is going to be the negative of that. So it's going to be equal to negative two. And so you could keep going with that. You could see that whatever f of a certain value is, g of that value would be the negative of that. So it would be down here. And so g of x would be a reflection of f of x about the x-axis. So g of x is going to look something like that, a reflection about the x-axis. Once again, on Khan Academy, you'd pick the choice that would actually look like that. Let's do another example. His is strangely fun. (laughs) All right, so here we're told functions f, so that's in solid in this blue color, and g dashed, so that's right over there, are graphed. What is the equation of g in terms of f? So pause this video and try to think about it. So the key is to realize how do we transform f of x, actually, they've labeled it over here, this is f of x right over here, in order to get g? So f of negative x would be a reflection of f about the y-axis. And so it would intersect there. It would have this straight portion like this. And I'm just experimenting right now. It'd have the straight portion like this. And then it would go up. And let's see. If f of negative x. So when you input six into it, that would be f of negative six, which is six. So it would go up there. So f of negative x would look something like this. Something like that. So the purple is f of negative x. Now, that doesn't quite get us to g, but it gets us a little bit closer 'cause it looks like if I were to take the reflection of f of negative x, f of negative x about the x-axis, it looks like I'm going to get to g. And so how do you reflect something about the x-axis? Well, we saw it in the example just now. You multiply the entire function by a negative. So we could say that g is equal to the negative of f of negative x. It's equal to the negative of this. So we're doing both reflections. We're flipping over the y-axis, and we're flipping over the x-axis to get to g. Let's do one more example. So once again, they've graphed f, they've graphed g, and they've said f is defined as this right over here. What is the equation of g? So they're not just asking it in terms of f. They just wanna know what is the equation of g? Pause this video and try to think about it. Well, you can see pretty clearly that this is a reflection across the y-axis. And a reflection across the y-axis, you can see pretty clearly, that g of x is equal to f of negative x. F of negative x. How do we know that? Well, for whenever we take f of x and we get that value, g at the negative of that value takes on the same function value, I guess I could say. Or another way to think about it is we could just pick this point, negative eight. F of negative eight is equal to a little over four, but g of eight is equal to a little over four, is equal to that same value. And so what is the equation of g? Well, we just have to rewrite this so that we can write it out as an equation. And so we could write out g of x is equal to. If I were to replace all of the x's here with a negative x, what would I get? I would get four times the square root of two minus. Instead of an x, I will have a negative x. And then the minus eight is outside of the radical. And so we would have g of x is equal to four times the square root of two plus x minus eight. And we're done.