When you have an equation such as $y=2x$ and you want to rewrite it using function notation, it is conventional to define the function using another letter ($f$ for example). Why is the function not defined using the same letter? Is it because there would be ambiguity about whether you are referring to the coordinate or the function when using that letter?
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It is convenient for functions of several variables, where we have $f(x,y)$. Writing $y=2x$ is nice for graphing, where we have a y-axis – David Raveh Jan 04 '23 at 21:07
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The premise is wrong: sometimes you do see, for example, $y$ and $y(x)$ used interchangeably. – Adam Rubinson Jan 04 '23 at 21:43
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You're right that writing $y=y(x)$ uses the same symbol for both a coordinate and a function (which are quite different), but some people do it anyway and expect readers to disambiguate using context. It's a stylistic preference.
Karl
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$y$ and $x$ are free variables, an equality relation between the free variables gives a graph on the $xy$-plane (not necessarily a graph of a function). Your notation suggests that it can always be solved for $y$ in terms of $x$ as a function, and that is not correct. An example is $y^2+x^2=1$.
poeplva19
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