Problem: Find all functions that satisfy for all reals .
Solution: Let’s start by plugging in something easy, like . We get
so . Hmm, make it a little more complicated then and try just . Then
But wait, we already know so we can just plug this in. Thus we have
We check by plugging each in:
That means only works. QED.
Comment: Functional equations are pretty strange as far as problems go. Basically you go around plugging random things in until you find something useful. Then you work it all out and you usually get an interesting result.
Practice Problem: (360 Mathematical Contests – 1.1.49) Find all polynomials with integral coefficients such that
for all real numbers .
Leave a Reply
You must be logged in to post a comment.