Question

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

Answer 1

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(x) = y = sin(x)` 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 = 1-7x^2 \iff y = \pm sqrt(1-7x^2)`

That `pm` sign means that, given a certain input `x`, the output can be either `sqrt(1-7x^2)` or `-sqrt(1-7x^2)`.

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