How do you decide whether the relation 7x^2+y^2=1 defines a function?

Jun 14, 2018

It doesn't because one input value yields more than one output value.

Explanation:

A function is simply a rule that states how the inputs and outputs must be related.

For example, $f \left(x\right) = y = \sin \left(x\right)$ means that the rule is that, for any input $x$, the output will be the sine of that input.

In your case, if we solve the expression for $y$ in order to highlight this "rule", we have

${y}^{2} = 1 - 7 {x}^{2} \setminus \iff y = \setminus \pm \sqrt{1 - 7 {x}^{2}}$

That $\pm$ sign means that, given a certain input $x$, the output can be either $\sqrt{1 - 7 {x}^{2}}$ or $- \sqrt{1 - 7 {x}^{2}}$.

Since it is not true that every input yields one and only one output, this is not a function.