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

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Answer 1 · 2018-06-14T13:00:01

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

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.

How do you decide whether the relation 7x^2+y^2=17x^2+y^2=1 defines a function?