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I know that if a variable $z = f(x,y)$, then $z$ or $f$ is a function of $x$ and $y$. Consider $f = xy^2+y=5.$ Clearly, $xy^2+y=5$ is a curve on the x-y plane. $y$ and $x$ are implicitly related, and we can also say that $y$ is a function of $x$. But can we say that $f$ is a function of $x$ and $y$?

Sasikuttan
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  • $y$ is only locally a function of $x$. for a given $x$, there might be $2$ values of $y$ that work and functions must be single valued. I don't understand your second question. $f(x,y)=xy^2+y$ is certainly a function of $x$ and $y$. – lulu Jan 17 '22 at 15:01
  • @lulu $f(x,y) = xy^2 + y$ is certainly a function of $x$ and $y$. But my question is $f = xy^2 + y = 5$ a function of $x$ and $y$. The value of $f$ is a constant here, and $x$ and $y$ can not take any random values in $R^2$. What will be the range and domain of '$f$' in this case? – Sasikuttan Jan 17 '22 at 15:07
  • Well, I don't know what that means unless you either mean $f(x,y)=xy^2+y$ or you mean the constant function $5$. – lulu Jan 17 '22 at 15:11
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    $f(x,y)=xy^2+y$ is a function, $f(x,y)=5$ is a function but $xy^2+y=5$ is an equation defining a set of points. In the same way $f(x)=x$ and $f(x)=0$ are functions but $x=0$ is a single point. – Paul Jan 17 '22 at 15:14
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    @Curiouserandcuriouser: "$f=xy^2+y=5$" might best be described as an abuse of notation. If you're trying to say "$xy^2+y=5$ represents a curve in the $xy$ plane, and I want to call this representation (and/or the curve) $f$", then it would be better to actually say that instead of using "$=$" two different ways in a single phrase. For alternative symbology, you can write, say, $f:; xy^2+y=5$. Note that you can define $f(x,y)=xy^2+y$ uncontroversially as a function, and then discuss $f(x,y)=5$ as a level curve (or level set) of that function. – Blue Jan 17 '22 at 15:15
  • @lulu Consider the implicit relation between x and y such that f(x,y) = c (a constant). Can 'f' be called a function here? – Sasikuttan Jan 17 '22 at 15:15
  • @Paul Thanks, but isn't x=0 a line? – Sasikuttan Jan 17 '22 at 15:22
  • That implicit (local) function is not $f$. – lulu Jan 17 '22 at 15:35
  • @Curiouserandcuriouser: If $x=0$ then $f(x)=0$ so it corresponds to the point $(0,0)$ in 2D. – Paul Jan 18 '22 at 07:26

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You are conflating two different uses for the "$=$" sign.

In, say, $$x+y = 5$$ you are asked to think about all the pairs $(x,y)$ for which that equation holds. They happen to form a line in the plane.

In $$ z = x+y $$ you are using the expression on the right to define a function of two variables whose value you name "$z$". You can then plot the graph of that function in three space.

The first equation defines a level curve of the function defined by the second equation.

Related: the definition of the word "equation" in math

Ethan Bolker
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