### Video Transcript

The graph of ๐ of ๐ฅ equals ๐ฅ minus two squared plus three for the left-closed, right-open interval from two to โ is shown. Which of the points ๐ด three, one; ๐ต seven, four; and ๐ถ nine, five lies on the graph of its inverse function ๐ inverse of ๐ฅ?

First, we should recall that if a function ๐ of ๐ฅ is invertible, so its inverse function exists, then the graph of ๐ฆ equals ๐ inverse of ๐ฅ is obtained by reflecting the graph of ๐ฆ equals ๐ of ๐ฅ across the line ๐ฆ equals ๐ฅ. This has the effect of switching the roles of ๐ฅ and ๐ฆ in the equation of the function. And hence it follows that any coordinate for an inverse function can be found by swapping the ๐ฅ- and ๐ฆ-coordinates of the corresponding point on the graph of the original function. So, for example, a point with coordinates ๐, ๐ on the graph of ๐ฆ equals ๐ of ๐ฅ gets mapped to the point ๐, ๐ on the graph of ๐ฆ equals ๐ inverse of ๐ฅ.

So to determine whether any of the points ๐ด, ๐ต, or ๐ถ lie on the graph of the inverse function ๐ฆ equals ๐ inverse of ๐ฅ, we need to swap each pair of coordinates around and determine which of these points lies on the graph of ๐ฆ equals ๐ of ๐ฅ. Swapping the ๐ฅ- and ๐ฆ-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 ๐ฆ equals ๐ of ๐ฅ.

The point with coordinates one, three is located here, which is not on the graph of ๐ฆ equals ๐ of ๐ฅ. The point with coordinates four, seven is located here, and this is on the graph of ๐ฆ equals ๐ of ๐ฅ. And finally, the point with coordinates five, nine is located here, which is not on the graph ๐ฆ equals ๐ of ๐ฅ. This means that point ๐ต is the only one of the three points that will lie on the graph of ๐ฆ equals ๐ inverse of ๐ฅ because if we swap the ๐ฅ- and ๐ฆ-values around, this is the only point that lies on the graph of ๐ฆ equals ๐ of ๐ฅ.

We can also sketch the line ๐ฆ equals ๐ inverse of ๐ฅ by reflecting ๐ฆ equals ๐ of ๐ฅ in the line ๐ฆ equals ๐ฅ. And if we do, we see that the point ๐ต, which has coordinates seven, four, does indeed lie on this line. Our answer then is that it is only point ๐ต the lies on the graph of the inverse function ๐ inverse of ๐ฅ.