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Multivariate function taking just one variable?

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Why in some books is there a multivariate function specifying just one variable when it is supposed to take two or more. For example:
why

f(x)= x²+y²

 

and not

 

f(x,y)=x²+y²

 

Should yield the corresponding value of z.

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It is common with a lot of mathematicians to use a different font/style to denote a tuple or vector, for example \( \mathbb x = (x, y) \) and then define the function as \(f(\mathbb x) = f(x, y) = x^2 + y^2\).

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