 One specialized type of transformation comes when we try to graph an inverse function. So far we've talked about reflections across the x-axis, across the y-axis, and across the origin. But there's no reason we couldn't use any line as our line of reflection. So for example, we might reflect a point p across the line x equals 3. So let's take our point and the line x equals 3. And remember for a reflection, how far in front of the mirror you are determines how far behind the mirror you appear to be. Now finding the algebraic equation for the reflected curves is generally complicated, except in one important case, a reflection across the line y equals x. So let's take a look at that. If x, y is a point on the original graph, and we want to reflect this across the line y equals x, well let's draw that line y equals x. Now the coordinates of our point are x, y, which means that we have a vertical distance y and a horizontal distance x. When we reflect this point across the line y equals x, we see these horizontal and vertical distances change places. So this horizontal distance is y, and this vertical distance is x. So the coordinates of this point, horizontal distance y, vertical distance x, so the coordinates are y, x. And so if x, y is a point on the original graph, a reflection across the line y equals x will move it to the point x, y, where x is the old y value, and y is the old x value. So suppose the graph of y equals x squared is reflected across the line y equals x. Let's find the equation of the new graph. So if x, y is a point on the original, then capital X, capital Y will be a point on the reflected graph where capital X is y, and capital Y is x. So from our equation y equals x squared equals, means replaceable will replace y with capital X, x with capital Y, and get our new equation. And so our reflected graph has the equation x equals y squared. Now let's think about what we have. Suppose we reflect the graph of y equals f of x across the line y equals x. If x, y is a point on the original, then y equals f of x. Now if capital X, capital Y is a point on the reflected graph, then since our reflection is going to involve changing x and y, capital X equals f of y. And so our reflected graph has equation x equals f of y. But according to our definition, if f of a equals b, then f inverse of b is equal to a. And so it appears that the reflection gives us the graph of y equals f inverse of x. So for example, suppose we have the graph of y equals f of x, we'll sketch the graph of y equals f inverse of x, and we should explain why our graph actually shows the inverse. So we'll start by reflecting this graph across the line y equals x. So now why does the reflected graph show the inverse? Let's take a look at that reflection again. Now consider any point x, y on the original graph, then y equals f of x. This point becomes y, x on the reflected graph. And what I'd like to do is I'd like to express our output, which is x in this case, in terms of our input, which is y in this case. So in this case, we have x equals g of y, or g is some function. But since y equals f of x, and x equals g of y, then g meets the definition of the inverse, so g is the inverse function. With these ideas in mind, we can lend some insight to the secret of the square root. Suppose y equals x squared. Then by our definition, x is a square root of y. We can try to find the inverse function. And we'll do that by reflecting the graph of y equals x squared across the line y equals x. But if we reflect y equals x squared across the line y equals x, the result is not a function of x. Because we see that in general any x value corresponds to two different values of y. Now if you imagine this curve to be like a tree or some sort of plant, we can imagine that there's two branches here. And so to solve this problem of not having a function, we need to pick a branch, a part of the curve, that is a function. And so for the square root, we'll prune the negative branch. We'll chop it off and get rid of it. And this gives us the graph of something that is a function and leads to our definition as square root of n using the symbol gives us the non-negative number whose square is n.