How to decide whether the relation 7x2+y2=1 defines a function?

Simply put, a function is a rule that specifies how inputs and outputs must relate to one another.
For example, ${\displaystyle f\left(x\right)=y=\sin \left(x\right)}$ means that the rule is that, for any input ${\displaystyle x}$, the output will be the sine of that input.
In your case, if we solve the expression for ${\displaystyle y}$ in order to highlight this ""rule"", we have${\displaystyle y^2=1-7x^2\iff y=\pm \sqrt{1-7x^2}}$
That ${\displaystyle \pm }$ sign means that, given a certain input ${\displaystyle x}$, the output can be either ${\displaystyle \sqrt{1-7x^2}}$ or ${\displaystyle -\sqrt{1-7x^2}}$.
This is not a function because it is not true that each input results in a single and unique output.