What is a self-inverse function? Take as an example . Substituting , we get , so the inverse function will map back onto 3. In this case, (and it does not happen very often), , and so it appears that the function acts as its own inverse. Of course, this needs to work for any starting value, which is not so easy to check. One way of expressing this is to say that , which can be shortened to .
The graph of an inverse function is the reflection of the graph of the function in the line . If a function is self-inverse, reflecting in this line must give the same graph again. This means that the graph of a self-inverse function must be symmetrical about the line . The diagram shows the graph of , and the line , and this symmetry property is obvious.
It turns out that it is enough just to check that the x- and y-intercepts have this symmetry. If they do, then so will the whole graph, and we shall know that the function is self-inverse. For our example: , and so the y-intercept is
, so the x-intercept is
Applying this idea to the general rational function, :
and , giving intercepts at and .
The function is self-inverse if , so the condition required is that .
Mathematicians love to generalise, and although there is no obvious application for what follows, it is a fascinating topic to explore. The function has the property that
If you are going to read any further, you must have a calculator to hand, and know how to use the ANS button. The above calculation can be done very quickly by entering the number , so that is the last answer, and then type and press the EXE or ENTER button three times. You should then repeat this using different starting values and satisfy yourself that the equation is true for any starting value . (This is almost the case, but if you are unlucky enough to get a division by zero at any stage, then of course an error will occur.)
If you have made that work, try , again with several different starting values (Hint:: a cycle of length four?).
And for a real surprise, try . Magic!