Functional Equation

[homeschool girl]

The problem:

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$f(x) f(y) = f(x + y) + xy$$for all real numbers $x$ and $y.$ Find all possible functions $f.$

Hint: The first strategy for attacking a functional equation is to substitute values. Combinations of 0s and 1s are usually helpful.


What I've done so far:

I started by trying $x=0, y=1$ and got

$f(0) f(1) = f(1)$

which implies that $f(0)=1$ and/or $f(1)=0$

if $f(1) = 0$, then i think you could write $f(n)$ as

$f(n-1)f(1)=f(n-1+1)+n-1,$

$f(n)=-n+1.$

I'm not sure if that's correct though, and I'm stuck on how to find the other possible functions.

 

[Steven G]

if you're not sure if f(x) = 1-x works, then test it!
f(x)f(y) = (1-x)(1-y) = 1-x-y +xy.
f(x+y) + xy = (1 - (x+y)) + xy

Think some more about the other case.