Can 'f' be called a function in the given problem?

Question

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 $y$?

Answer 1

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