### Video Transcript

The following is the graph of π of π₯ equals two π₯ minus one. Which one is the graph of the inverse function, the inverse of π of π₯?

In order to answer this question, letβs remind ourselves what we mean by the inverse function. Suppose we have a function π of π₯. The inverse π denoted by the superscript negative one of π of π₯ is simply equal to π₯ for all values of π₯ in the domain of the original function. In other words, the inverse function undoes the original function. And so one technique we have to identify the inverse function given a graph is to begin by identifying its equation.

Now itβs outside the scope of this video to look at real formal methods. But we know the function π of π₯ takes a value of π₯, it timeses it by two, and then it subtracts one. The inverse function must undo this. So working backwards, the inverse function will add one, and then we will need to divide by two. This undoes the original function. Another way of representing this is as a half π₯ plus one-half.

We know that this represents a graph of a straight line. It has a slope of one-half, and it passes through the π¦-axis at one-half. So with this in mind, letβs plot the line π¦ is equal to the inverse of π of π₯ on our original axes. It has a π¦-intercept of one-half. In other words, it passes through the point zero, one-half. And it has a slope of positive one-half. So for every one unit right, which here is two small squares, we must move half a unit up. Thatβs one small square. And so there is the function π¦ is equal to the inverse π of π₯.

But of course, there is an easier way. The graph of the original function and the graph of the inverse function have switched values for π₯ and π¦. For instance, the original function passes through the point with coordinates two, three, whilst the inverse passes through the point with coordinates three, two. This corresponds to a reflection of the other graph in the line π¦ equals π₯. And in general, that is how we can find the graph of the inverse function. We reflect the original function in the line π¦ equals π₯.

Of course, either technique is entirely suitable here. And when we compare this graph with the four graphs weβve been given, we see this corresponds to graph (A). So given the function π of π₯ equals two π₯ minus one, the graph of the inverse function is (A).