zmudz one year ago 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!

1. anonymous

Have you tried it yet?

2. freckles

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.

3. freckles

yep it looks like that works