Question Video: Determining the Coordinates of a Point That Lies on the Graph of an Inverse Function Given the Graph of the Function Itself Mathematics

Video Transcript

The graph of 𝑓 of 𝑥 equals 𝑥 minus two squared plus three for the left-closed, right-open interval from two to ∞ is shown. Which of the points 𝐴 three, one; 𝐵 seven, four; and 𝐶 nine, five lies on the graph of its inverse function 𝑓 inverse of 𝑥?

First, we should recall that if a function 𝑓 of 𝑥 is invertible, so its inverse function exists, then the graph of 𝑦 equals 𝑓 inverse of 𝑥 is obtained by reflecting the graph of 𝑦 equals 𝑓 of 𝑥 across the line 𝑦 equals 𝑥. This has the effect of switching the roles of 𝑥 and 𝑦 in the equation of the function. And hence it follows that any coordinate for an inverse function can be found by swapping the 𝑥- and 𝑦-coordinates of the corresponding point on the graph of the original function. So, for example, a point with coordinates 𝑎, 𝑏 on the graph of 𝑦 equals 𝑓 of 𝑥 gets mapped to the point 𝑏, 𝑎 on the graph of 𝑦 equals 𝑓 inverse of 𝑥.

So to determine whether any of the points 𝐴, 𝐵, or 𝐶 lie on the graph of the inverse function 𝑦 equals 𝑓 inverse of 𝑥, we need to swap each pair of coordinates around and determine which of these points lies on the graph of 𝑦 equals 𝑓 of 𝑥. Swapping the 𝑥- and 𝑦-values around for each pair of coordinates gives the points one, three; four, seven; and five, nine. We can then plot each of these points on the coordinate grid given and identify which, if any, of the three points lie on the graph of 𝑦 equals 𝑓 of 𝑥.

The point with coordinates one, three is located here, which is not on the graph of 𝑦 equals 𝑓 of 𝑥. The point with coordinates four, seven is located here, and this is on the graph of 𝑦 equals 𝑓 of 𝑥. And finally, the point with coordinates five, nine is located here, which is not on the graph 𝑦 equals 𝑓 of 𝑥. This means that point 𝐵 is the only one of the three points that will lie on the graph of 𝑦 equals 𝑓 inverse of 𝑥 because if we swap the 𝑥- and 𝑦-values around, this is the only point that lies on the graph of 𝑦 equals 𝑓 of 𝑥.

We can also sketch the line 𝑦 equals 𝑓 inverse of 𝑥 by reflecting 𝑦 equals 𝑓 of 𝑥 in the line 𝑦 equals 𝑥. And if we do, we see that the point 𝐵, which has coordinates seven, four, does indeed lie on this line. Our answer then is that it is only point 𝐵 the lies on the graph of the inverse function 𝑓 inverse of 𝑥.