Insight on symmetrical property of function inverse

Aug 5 2024

Intersection between a function and its inverse

A past H2 Mathematics examination question 2008 P2 Q4iii asked for the exact solution to the equation f(x)=f−1(x). Getting an exact solution required insight on the symmetrical property of the graph of y=f(x) and y=f−1(x) about the line y=x.

And for the longest time, that was a trick I often employed to solve questions like this: to solve f(x)=f−1(x), we solve f(x)=x instead.

This trick may not work all the time

A question from EJC 2023 Prelims presented a situation where this trick will not work. They used a more complicated function, but I could replicate similar results with a function like f(x)=−x3. Turns there are three roots to the equation f(x)=f−1(x) for this example, while the equation f(x)=x has only one root.

This led to me questioning myself if I have left out answers whenever I employed the trick in the past. It's time to investigate this further using the mathematician's favorite tool: proofs.

Formulating the problem

We consider only numbers that are in the domains Df∩Df−1. Let A denote the set of numbers that satisfy the equation f(x)=x and let B denote the set of numbers that satisfy the equation f(x)=f−1(x).

We wish to understand the relationship between sets A and B. As we have seen in the previous example of f(x)=−x3, the two sets need not be the same.

The result

  1. A⊆B.
  2. If f is strictly increasing, then A=B.

The proof

Proof of 1

Let x∈A. Then we have f(x)=x. Since we restrict our universal set to only numbers in Df∩Df−1, we can apply the inverse function on both sides to get f−1f(x)=f−1(x) which simplifies to x=f−1(x). Thus we have x=f(x)=f−1(x) so x∈B.■

Proof of 2

While the first proof came intuitively to me, I was not able to figure out the "strictly increasing" condition necessary for the two sets to be equal so I did some googling. I found the result I was looking for, and its proof, in vaster's answer to a question on Math StackExchange.

Reflecting on the beauty of mathematical proofs

I look back fondly on my time in university. Learning about the various mathematical results and/or proofs is often a joy, and especially when a proof is especially "elegant" and enlightening. Erdős's reference to "The Book" is a imagery I often have after digesting a good proof.

It is unfortunate I do not encounter such insights as often as I have ventured into my career teaching pre-university mathematics.

The proof is probably too trivial to enter "The Book", but I found it especially enlightening nevertheless. It showcases why both components of the condition (strictly and increasing) are necessary to get our desired result.

The proof by contradiction is a very standard technique, but employing the law of trichotomy makes everything click into place. The increasing condition is needed to preserve order, while the strictly increasing condition is needed to arrive at our contradiction.