Answer
No, this relation is not a function, because it maps every value of $x$ in the domain to two values of $y$ in the codomain. This is evidenced by the fact that the graph of the function fails the vertical line test.
Work Step by Step
Note, for example, that if we let $x=4=y^{2}$, then either $y=2$ or $y=-2$. A function must map every value in the domain onto exactly one value in the codomain.
