Kamran011 wrote: Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$for all $x, y\in \mathbb{R^{+}}$ Let $P(x,y)$ be the assertion $f(x)^2=f(xy)+f(x+f(y))-1$ $P(1,x)$ $\implies$ $1+f(1)^2=f(x)+f(1+f(x))$ and so $f(x)< 1+f(1)^2$ $\forall x$ and so $f(x)$ is upperbounded. Let $M=\sup_{x>0}f(x)$ Let $x_n$ a sequence of positive reals such that $\lim_{n\to+\infty}f(x_n)=M$ (which exists, by definition of $M$) $P(x_n,\frac x{x_n})$ $\implies$ $f(x)=f(x_n)^2-f(x+f(\frac x{x_n}))+1$ And so $M\ge f(x)\ge f(x_n)^2-M+1$ Setting there $n\to+\infty$, we get $M\ge f(x)\ge M^2-M+1$ And so $M=1$ and $\boxed{f(x)=1\quad\forall x}$, which indeed fits.