14. Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that $x f(y)=y f(x)$ for all $x, y \in \mathbb{R}$. Find the intersection of the graphs of $y=f(x)$ and $y=x^{2}+1$ if $f(1)=-1$.
(ans $\phi=$ Null set.
We have that $\frac{f(x)}{x}=\frac{f(y)}{y}=c$, a constant $\Rightarrow f(x)=c x \Rightarrow f(x)=$ $-x$ from the given condition. $y=f(x)=-x$ does not intersect the parabola $y=x^{2}+1$ because $x^{2}-x+1=0$ has no real solutions.)