Question

The function \(f : \mathbb{R} \rightarrow \mathbb{R}\) satisfies \(xf(x) + f(1 - x) = x^3 - x\) for all real \(x\). Find \(f(x)\).

This is for precalc, any help is appreciated!

Answer 1

Have you tried it yet?

Answer 2

let u=1-x
so
\[\text{ if } u=1-x \text{ then } x=1-u \\ \text{ so we have } \\ (1-u)f(1-u)+f(u)=(1-u)^3-(1-u) \\ \text{ so you have a system \to solve } \\ (1-x)f(1-x)+f(x)=(1-x)^3-(1-x) \\ xf(x)+f(1-x)=x^3-x\]
I think this might work
I haven't tried to solve the system yet myself.

Answer 3

yep it looks like that works