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

The graph of ๐‘“(๐‘ฅ) = (๐‘ฅ โˆ’ 2)ยฒ + 3 for the interval [2, โˆž) is shown. Which of the points ๐ด (3, 1), ๐ต (7, 4), and ๐ถ (9, 5) lies on the graph of its inverse function ๐‘“โปยน(๐‘ฅ)?

02:43

Video Transcript

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

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

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

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

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

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