The equation $y^{2}=x$ defines a parabola. This curve is not the graph of a function of $x$. Why not?

Solution. The value $x$ is associated with two values $y=1$ and $y=-1$.

Not all curves are graphs of functions.

The positive square root gives a function. Let the domain be $A=\{x\in{\mathbb{R}}:x\geq 0\}$ and the codomain be $B=\{x\in{\mathbb{R}}:x\geq 0\}$, and take $f:A\rightarrow B$ to be $f(x)=\sqrt{x}.$