Question Video: Comparing the Graph of a Function With Its Inverse Mathematics • Second Year of Secondary School

Video Transcript

If we drew the function 𝑓 of 𝑥 and its inverse function, the inverse of 𝑓 of 𝑥, on the same graph, which of the following must be true? (A) There would be symmetry about the line 𝑦 equals 𝑥. (B) There would be symmetry about the 𝑥-axis. (C) There would be symmetry about the origin. (D) There would be symmetry about the 𝑦-axis. Or (E) the two graphs would be the same.

In order to answer this question, let’s remind ourselves of the links between the function 𝑓 of 𝑥 and its inverse. Suppose we have the function 𝑓 whose inverse is given by 𝑓 with a superscript negative one. The inverse of 𝑓 of 𝑥 is simply equal to 𝑥 for all 𝑥 in the domain of the function. In other words, the inverse of the function actually undoes the original function. And there are a number of techniques we can use to find the inverse of a function. It’s somewhat outside the scope of this video to demonstrate any of those. But we will look at a very quick example.

Suppose we have the function 𝑓 of 𝑥 equals two 𝑥. This function takes any values in the domain and multiplies them by two. The inverse of 𝑓 is the function that undoes this process. So it’s the function that takes values and divides them by two. The inverse of 𝑓 in this case then is 𝑥 divided by two. We’ll demonstrate what happens graphically by plotting each of these in the first quadrant of the coordinate plane. Let’s begin with 𝑦 equals 𝑓 of 𝑥, the graph of 𝑦 equals two 𝑥. This has a 𝑦-intercept of zero. So it passes through the origin or the point zero, zero. It also has a slope of two. This means it will pass through the point with coordinates one, two and two, four.

In a similar way, the graph of the inverse 𝑦 equals 𝑥 over two also passes through the origin. This time though, it has a slope of one-half. And so it passes through the point two, one and the point four, two. If we look very carefully, we can see that we can map either of these graphs onto the other by reflection across the line 𝑦 equals 𝑥. And this makes a lot of sense, since we’re essentially swapping the 𝑥- and 𝑦-values of each coordinate. So the point with coordinates one, two maps onto two, one and vice versa. Similarly, the point with coordinates two, four maps onto four, two and vice versa.

This means if we draw the function 𝑓 of 𝑥 and its inverse on the same graph, we know they are symmetrical across the line 𝑦 equals 𝑥. And so the answer is (A) there would be symmetry about the line 𝑦 equals 𝑥.