Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

Question

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that there are more solutions but I don't know how to find them.

Answer 1

HINT:

As $f(x)-x=f(1-x)-(1-x),$ put $f(x)-x=g(x)$

I'm tempted to add this :

(As IvanLoh has pointed out), if we need $f(x)$ in polynomials

As $f(x)-f(1-x)=2x-1$ which is $O(x^1), f(x)$ can be at most Quadratic

Let $f(x)=ax^2+bx+c$

$\implies 2x-1=f(x)-f(1-x)=ax^2+bx+c-\{a(1-x)^2+b(1-x)+c\}$ $\implies 2x-1=-(a+b)+2(a+b)x^2$

Equating the constants $a+b=1$

and equating the coefficients of $x,a+b=1\implies b=1-a$

So, any $f(x)=ax^2+(1-a)x+c$ will satisfy the given condition

Answer 2

Hint: Let $f(x)=x^2+g(x)$. What properties must $g(x)$ have?