## Tuesday, August 22, 2006

### Function + Equation = ?? Topic: Algebra. Level: AIME/Olympiad.

Problem: Find all functions $f(x)$ that satisfy $f(x)f(y)-f(xy) = x+y$ for all reals $x,y$.

Solution: Let's start by plugging in something easy, like $x = y = 1$. We get

$[f(1)]^2-f(1) = 2$

$[f(1)-2][f(1)+1] = 0$

so $f(1) = 2, -1$. Hmm, make it a little more complicated then and try just $y = 1$. Then

$f(x)f(1)-f(x) = x+1$

$f(x)[f(1)-1] = x+1 \Rightarrow f(x) = \frac{x+1}{f(1)-1}$.

But wait, we already know $f(1) = 2, -1$ so we can just plug this in. Thus we have

$f(x) = x+1$ or $f(x) = -\frac{x+1}{2}$.

We check by plugging each in:

$f(x)f(y)-f(xy) = (x+1)(y+1)-(xy+1) = x+y$ (this works)

$f(x)f(y)-f(xy) = \left(-\frac{x+1}{2}\right) \left(-\frac{y+1}{2}\right) - \left(-\frac{xy+1}{2} \right) \neq x+y$ (this doesn't).

That means only $f(x) = x+1$ works. QED.

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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.

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Practice Problem: (360 Mathematical Contests - 1.1.49) Find all polynomials $P(x)$ with integral coefficients such that

$P(P^{\prime}(x)) = P^{\prime}(P(x))$

for all real numbers $x$.