Find all f such that $f(f(y))+f(x-y)=f(xf(y)-x)$

Question

Find all functions $f$ defined over real numbers to real numbers such that $f(f(y))+f(x-y)=f(xf(y)-x)$

My attempt: Set $x=y=0$ to get $f(f(0))=0$. It will be very helpful if I will able to find $f(0)$ but I failed to find it. I tried to check the injectivity of $f$ but wasn't able to check it.

Please help me.

Answer 1

I tried to solve this problem with taking help from the above comments.Please check it.

I am tooo lazy to LaTeX it .Please don't mind!

Answer 2

Let $P(x,y)$ be the assertion that $f(f(y))+f(x-y)=f(xf(y)-x)$. Now,

Observation 1: $P(0,0)\implies f(f(0))=0$.

Observation 2: Considering $P(0,f(0))$ and $P(f(0),f(0))$ we get $f(0)=0$.

Observation 3: Considering $P(x,0)$ we get $f$ is even function, i.e;$f(x)=f(-x)$ and finally $P(x,-x)$ and $P(x,x)$ implies $f(x)=0$.

So, $f(x)=0,\forall x\in\mathbb{R}$.