algartaim6jf

2023-01-01

How to decide whether the relation $7{x}^{2}+{y}^{2}=1$ defines a function?

Giaj5sf

Expert

Simply put, a function is a rule that specifies how inputs and outputs must relate to one another.
For example, $f\left(x\right)=y=\mathrm{sin}\left(x\right)$ 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${y}^{2}=1-7{x}^{2}⇔y=±\sqrt{1-7{x}^{2}}$
That $±$ sign means that, given a certain input $x$, the output can be either $\sqrt{1-7{x}^{2}}$ or $-\sqrt{1-7{x}^{2}}$.
This is not a function because it is not true that each input results in a single and unique output.

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