Mathematics

Solving a Simple Functional Equation

This functional equation looks quite similar to Cachy’s functional equation we discussed earlier, so a natural question arises — can we apply it to this problem? Spoiler: the answer is a Yes!

As always, try to solve this problem yourself before reading the solution. If your goal is to become better at math, simply reading the solutions won’t help you much.

Solution

First, as always, try to plug in some numbers (though this might not be too helpful in this particular problem) and find possible solutions to this equation. After some time, you might find that

is a solution. But how do we find other solutions or prove they don’t exist? The trick is to define

Now if we plug that into the original equation we will get

And this is exactly the Cauchy’s functional equation! Immediately we get

and thus

Finally, we check that this solution works, and we are done.

Exercises

In case you want to try to solve similar problems yourself, I include a list of exercises:

Credit: Ozgur Kircak

Have fun solving them!

Exploring Cauchy’s functional equation