Let $P(x,y)$ be the assertion. First,it is easy to see that $\boxed{f(x)\equiv 0}$ is a solution. Now,we will assume that $f(x)\not\equiv 0$. $P(1,1)\implies f(1)f(f(1)-1)=0\implies \exists\, a\in\mathbb{R},f(a)=0$. $P(a,0)\implies a^2f(0)=0\implies f(0)=0$. $P(x,0)\implies f(x)f(-1)=-f(x)\implies f(-1)=-1$. $P(-1,-y)\implies f(y-1)=-f(-y)-1^{(*)}$. Hence the assertion become $P(x,y);f(x)[-f(-yf(x))-1]=x^2f(y)-f(x)\implies \boxed{f(x)f(-yf(x))=-x^2f(y)}$. If $f(d)=0$,then $P(d,y)\implies d=0$.Hence $d\neq 0\implies f(d)\neq 0$. Now,for any $x\neq 0$,we have $P(x,x)\implies f(-xf(x))=-x^2$.Hence all nonpositive real numbers are in the range of $f(x)$. From the new assertion,it is easy to see that $f(a)=f(b)\implies a=\pm b$. But if $f(a)=f(-a)\,\exists a\neq 0$,then $(*)\implies f(a-1)=f(-a-1)\implies a-1=\pm (-a-1)$ which is a contradiction.Hence $f(x)$ is injective. Thus $P(x,x)\implies f(-xf(x))=-x^2\,\forall x\neq 0\implies f(-xf(x))=f(xf(-x))=-x^2\implies -f(x)=f(-x)\,\forall x\neq 0$. Hence $f(x)$ is surjective and $f(y-1)=f(y)-1$. Thus the assertion now become $P(x,y);\boxed{f(x)f(yf(x))=x^2f(y)}$. $P(x,1)\implies f(x)f(f(x))=x^2f(1)=-x^2f(-1)=x^2$. Hence $f(x)f(yf(x))=f(x)f(f(x))f(y)\implies f(yf(x))=f(f(x))f(y)\,\forall x\neq 0\implies f(yf(x))=f(f(x))f(y)\,\forall x,y\in\mathbb{R}$. But since $f(x)$ is surjective,we get that $f(x)$ is multiplicative. $P(x,y+1)-P(x,y)\implies f(x)[f(yf(x)+f(x))-f(yf(x))]=x^2=f(x)f(f(x))\implies f(z+a)-f(z)=f(a)\,\forall a\neq 0$. Now it is easy to see that $f(x)$ is addictive,too.Hence $\boxed{f(x)\equiv x}$.(Since we have exclude $f(x)\equiv 0$.) Hence all solution are $f(x)\equiv 0$ and $f(x)\equiv x$ which clearly satisfied the assertion.

AnArtist wrote: Claim 2) $f(\frac{1}{x})= \frac{f(x)}{x^2}$ $\forall x \neq 0$. Proof $P( x, \frac{1}{f(x)})$. $P(x,\frac {1}{f(x)}) $ gives $f(\frac {1}{f(x)})=\frac {f(x)}{x^2} $

Here is a bit of improvement on the solutions above. If $f$ is non-constant, $P(0,y): f(0)=0$. If $f(c)=0, P(c,y): c=0$ $P(1,1): f(1)f(f(1)-1)=0$ Thus, $f(1)=1$. $P(1,y): f(y+1)=f(y)+1$ Hence the original equation becomes $f(x)f(yf(x))=x^2f(y)$ We get $f(x)f(f(x))=x^2$ and $f(x+1)=f(x)+1$, we can also do: $x^2+2x+1=(x+1)^2=f(x+1)f(f(x+1))=(f(x)+1)(f(f(x))+1)=f(x)f(f(x))+f(x)+f(f(x))+1=x^2+f(x)+f(f(x))+1$ And so $f(x)+f(f(x))=2x$. Lwt $f(x)=a, f(f(x))=b$, $ab=x^2, a+b=2x$ so $b=2x-a \implies a(2x-a)=x^2 \implies (x-a)^2=0$ Thus, $f(x)=a=x$

dominicleejun wrote: Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers, such that $f(x)f(yf(x) - 1) = x^2 f(y) - f(x) \quad\forall x,y \in \mathbb{R}$ let $p(x,y)=f(x)f(yf(x)-1)=x^2f(y)-f(x)$ clearly $p(0,y)$ either gives $f(x)=0$ or $f(0)=0$ so assume $f(0)=0$ then $p(x,0)=f(x)f(-1)=-f(x)$ so since $f(x)$ is not the zero constent $f(-1)=-1$ now $p(-1,y)=f(-y-1)=-f(y)-1$ so $f(y-1)=-f(-y)-1$ so $f(yf(x)-1)=-f(-yf(x))-1$ so rewrite as $p(x,y)=-f(x)f(-yf(x))=x^2f(y)$. now $p(x,-1)=f(x)f(f(x))=x^2$. now if $f(x)=0$ then $p(x,y)=x^2f(y)=0$ so $x=0$. now rewrite as $p(x,y)=f(-yf(x))=-f(y)f(f(x))$ now if $f(a)=f(b)$ then $a^2=b^2$ also as we had $f(-y-1)=-f(y)-1$ we have $f(-a-1)=f(-b-1)$ so $(b+1)^2=(a+1)^2$ which gives us $a=b$. now $p(x,x)=f(-xf(x))=-x^2=p(-x,-x)$ so $-xf(x)=xf(-x)$ which gives $-f(x)=f(-x)$ so rewrite equations as $f(y+1)=f(y)+1$ $p(x,y)=f(yf(x))=f(y)f(f(x))$ as $x \in D_f$ for $x<0$ and clearly if $a \in D_f$ then $-a \in D_f$ we have $f$ is surgective. so rewrite as $p(x,y)=f(xy)=f(x)f(y)$ so $f((x+1)y)=f(x+1)f(y)=f(x)f(y)+f(y)=f(xy)+f(y)$ so $f(a+b)=f(a)+f(b)$ . these give $f(x)=x$ for all real $x$.

Let $P(x,y)$ be the assertion $f(x)f(yf(x)-1)=x^2f(y)-f(x)$. $P(1,1)\Rightarrow f(1)f(f(1)-1)=0$ If $f(1)=0$, then $P(1,x)\Rightarrow\boxed{f(x)=0}$. Otherwise, there is a $j$ with $f(j)\ne0$ and $f(f(1)-1)=0$. $P(f(1)-1,j)\Rightarrow f(1)=1$ $P(1,x+1)\Rightarrow f(x+1)=f(x)+1$ $P(x,1)\Rightarrow f(x)f(f(x))=x^2$ Taking $x\mapsto x+1$ in this, we see that $f(x)+f(f(x))=2x$. Then $f(x)(2x-f(x))=x^2$, so $\boxed{f(x)=x}$.

As per usual, we use $P(x,y).$ Assume not $f\equiv 0$ (which works.) We get $f(1)(f(1)-1)=0$ from $P(1,1).$ Take $f(z)=0,$ then $P(z,y)$ implies $z=0.$ In particular $f(1)=1.$ \begin{align*} P(1,x)\implies &f(x-1)=f(x)-1 \\ P(a,b)\implies &f(a)f(bf(a))=a^2f(b) \\ P(a,1)\implies & f(a)f(f(a))=a^2f(1) \\ P(a+1,1)\implies & f(f(a))=2a-f(a) \\ P(a,1)\implies & f(a)\equiv a \quad \text{which fits.} \end{align*}

Let $P(x,y)$ denote the given assertion. $P(0,x): f(0)f(xf(0)-1)=-f(0)$. So either $f(0)=0$ or $f(xf(0)-1)=-1$ for any $x$. If $f(0)\ne 0$, then $xf(0)-1$ can take on any real value, which implies $f\equiv -1$, which is not a solution. Thus, we have $f(0)=0$. Now, noting $\boxed{f\equiv 0}$ works, we can assume $f$ is non-constant. Claim: $f$ is injective. Proof: Suppose $f(a)=f(b)$ with $a\ne b$. $P(a,x): f(a)f(xf(a)-1)=a^2f(x)-f(a)$. $P(b,x): f(b)f(xf(b)-1)=b^2f(x)-f(b)$. This implies $a^2f(x)=b^2f(x)$. If we set $x$ such that $f(x)\ne 0$, then $a^2=b^2\implies a=\pm b\implies a=-b$, since $a\ne b$. Then $f(a)=f(-a)$. In fact, this implies $f$ is injective at $0$. $P(a,-a): f(a)f(-af(a)-1)=a^2f(a)-f(a)$. $P(-a,a): f(a)f(af(a)-1)=a^2f(a)-f(a)$. If $f(a)\ne 0$, then we have $f(-af(a)-1)=f(af(a)-1)$. However, this implies either $af(a)+1=af(a)-1$, or $-af(a)-1=af(a)-1$, both are absurd. So $f(a)\ne f(-a)$. If $f(a)=0$, then $a=0$. Since $f$ is injective at $0$, $f$ is injective. $\blacksquare$ $P(x,0): f(x)f(-1)=-f(x)$. If we set $x\ne 0$, then we get $f(-1)=-1$. $P(1,1): f(1)f(f(1)-1)=0$, so $f(1)=1$. $P(1,x): f(x-1)=f(x)-1\implies f(x+1)=f(x)+1$. Now we rearrange the FE. We have $f(x)(f(yf(x))-1)=x^2f(y)-f(x)$, so\[f(x)f(yf(x))=x^2f(y)\]Let $Q(x,y)$ be the assertion here. $Q(x,1): f(x)f(f(x))=x^2\implies f(f(x))=\frac{x^2}{f(x)}$. $P(x+1,1): (f(x)+1)f(f(x))=(x+1)^2-f(x)-1=x^2+2x-f(x)\implies f(f(x))=\frac{x^2+2x-f(x)}{f(x)+1}$. This implies\begin{align*} \frac{x^2}{f(x)}=\frac{x^2+2x-f(x)}{f(x)+1} \\ \implies x^2(f(x)+1)=f(x)(x^2+2x-f(x)) \\ \implies x^2f(x)+x^2=x^2f(x)+2xf(x)-f(x)^2 \\ \implies x^2=2xf(x)-f(x)^2 \\ \implies f(x)^2-2xf(x)+x^2=0 \\ \implies (f(x)-x)^2=0 \\ \implies \boxed{f(x)=x} \\ \end{align*}which clearly works.

The answers are \( f \equiv 0 \) and \( f \equiv x \), which clearly work. Let \( P(x, y) \) denote the assertion. We start by making the following claim: Claim: For all \( x \in \mathbb{R} \), we have \[ f(x)f(f(x)) = x^2. \] Proof We first note that \( P(x, 0) \) gives us \[ f(x)f(-1) = -f(x) \implies f(x)(f(-1) + 1) = 0. \]So either \( f(x) = 0 \) for all \( x \in \mathbb{R} \) or \( f(-1) = -1 \). Note that \( f \equiv 0 \) is indeed a solution, so assume \( f(-1) = -1 \). Now we observe that: \(\bullet\) \( P(-1, y) \implies f(-y - 1) = -(f(y) + 1) \). \(\bullet\) \( P(x, -1) \implies f(x)f(-f(x) - 1) = -x^2 - f(x) \). Comparing the first and second bullets, we get that: \[ f(x)[f(f(x)) + 1] = x^2 + f(x) \implies f(x)f(f(x)) = x^2, \]as claimed. Now, we show that if there exists a \( z \neq 0 \) such that \( f(z) = 0 \), then \( f \) is zero everywhere. Indeed, by \( P(z, x) \), \[ z^2f(x) = 0 \implies f(x) = 0, \]as needed. Now \( P(0, x) \) gives us that \[ f(0)f(xf(0) - 1) = -f(0) \implies f(0)[f(f(0)x - 1) + 1] = 0. \]If \( f(0) \neq 0 \), then \( f(x) = -1 \) for all \( x \), which is not a solution. So indeed \( f(0) = 0 \). Thus we can write \[ f(f(x)) = \frac{x^2}{f(x)}, \]for all \( x \neq 0 \). We assume \( x \neq 0 \) from now on. We also realize that \( P(f(x), f(x)) \) gives us \[ f(f(x))f(f(f(x))f(x) - 1) = f(x)^2f(f(x)) - f(f(x)), \]\[ \frac{x^2}{f(x)}f(x^2 - 1) = x^2f(x) - \frac{x^2}{f(x)}, \]\[ f(x^2 - 1) = f(x)^2 - 1. \] This gives us \( f(1) = \pm 1 \). Note that \( f(1) = -1 \) does not work as \( P(-1, 1) \) gives \( f(-2) = 0 \). So \( f(1) = 1 \). Now, \( P(1, x) \) gives us that \[ f(x - 1) = f(x) - 1 \]for all natural \( n \). Therefore, the equation becomes \[ f(x)f(yf(x)) = x^2f(y), \]which we now denote by \( P(x, y) \). Now, \( P(x, x) \) gives \[ f(xf(x)) = x^2. \]Finally, \( P(xf(x), y) \) implies \[ f(x^2y) = f(x)^2f(y). \]Note that \( y = 1 \) gives \[ f(x^2) = f(x)^2, \]so the equation becomes \[ f(x^2y) = f(x^2)f(y). \]Holding \( y \) non-negative, we get that \( f \) is multiplicative in \( \mathbb{R}^+ \), but as \( f \) is odd, we have \( f \) is multiplicative everywhere. To finish, as we have \[ f(x + 1) = f(x) + 1, \]let \( x \rightarrow x/y \) for \( y \neq 0 \), you get \[ f(x + y)f\left(\frac{1}{y}\right) = f(x)f\left(\frac{1}{y}\right) + 1, \]\[ f(x + y) = f(x) + f(y), \]as \( f(1/y) = 1/f(y) \). Therefore, \( f \) is additive as well. But as \( f(x^2) = f(x)^2 \geq 0 \), by Cauchy, we have \( f(x) = cx \) for all \( x \in \mathbb{R} \). Plugging it back in, we get \( c \in \{0, 1\} \), giving us the solutions.

\[f(x)(f(yf(x) - 1)+1)=x^2f(y)\]Answers are $f\equiv 0$ and $f(x)=x$. Assume that $f$ is non-constant. Plugging $x=0$ yields $f(0)(f(yf(0)-1)+1)=0$. $f(0)\neq 0$ contradicts with our assumption hence $f(0)=0$. $y=0$ implies $f(x)f(-1)+f(x)=0$ thus, $f(-1)=-1$. Note that if $f(a)=0$, then $x=a$ gives $a^2f(y)=0$ which is impossible for $a\neq 0$. $x=y$ implies $f(xf(x)-1)=x^2-1$ which gives that $f$ is surjective over positive reals. Also for $x=1$, $f(f(1)-1)=0$ or $f(1)=1$. Plugging $x,f(y)$, by symmetry we get $\frac{x^2f(f(y))}{f(x)}=\frac{y^2f(f(x))}{f(y)}$ thus, $\frac{f(f(x))f(x)}{x^2}$ is constant. For $x=1$, it's equal to $1$ hence $x^2=f(x)f(f(x))$. Plug $y=1$ to see that \[f(x)(f(f(x)-1)+1)=x^2\iff f(f(x)-1)=f(f(x))-1\overset{\text{surjectivity for positive reals}}{\implies} f(x^2-1)=f(x^2)-1\]$P(-1,-y^2)$ gives $-f(y^2)=-f(y^2-1)-1=f(-y^2)$ hence $f$ is odd which proves the surjectivity for all reals. Now we can choose $f(x)$ as any real on equation $f(f(x)-1)=f(f(x))-1$ so $f(x-1)=f(x)-1\iff f(x+n)=f(x)+n$. \[f(x)f(yf(x))=x^2f(y)\]Compare $P(x,y)$ with $P(x,y+1)$ to observe $f(x)f(yf(x)+f(x))=x^2f(y)+x^2=f(x)f(yf(x))+x^2$. After plugging $x,\frac{y}{f(x)}$ we get $f(x)(f(y+f(x))-f(y))=x^2$ or $f(y+f(x))-f(y)=f(f(x))$ Since $f$ is surjective, $f(x+y)=f(x)+f(y)$ hence $f$ is additive. \[(f(x)+f(z))(f(yf(x))+f(yf(z)))=(x+z)^2f(y)\iff 2xzf(y)=f(x)f(yf(z))+f(z)f(yf(x))\]Plug $y=z=1$ to verify $2x=f(x)+f(f(x))$. Also $f(x)f(f(x))=x^2$ hence $x^2=f(x)(2x-f(x))\iff (x-f(x))^2=0$ which implies $f(x)=x$ for all reals as desired.$\blacksquare$