Proving the function with this property is bijective

Question

I do not know how to get $f(x)$, so I can see if it is bijective. $$f\colon \mathbb{R}\to \mathbb{R}$$ $$2f(3-2x)+f(2x-2)=x.$$

Answer 1

If we substitute $u:= 2x-2$, then the functional equation becomes $$2f(1-u)+f(u)=\frac{u+2}{2}.\tag{1}\label{1}$$ Evaluating at $1-u$ gives $$2f(u)+f(1-u)=\frac{1-u+2}{2}=\frac{3-u}{2}.\tag{2}\label{2}$$ Subtracting \eqref{2} twice from \eqref{1} gives $$-3f(u)=\left[2f(1-u)+f(u)\right]-2\left[2f(u)+f(1-u)\right]=\frac{3u-4}{2},$$ so that $$f(u)=-\frac{u}{2}+\frac{2}{3},$$ which is clearly a bijective function.