A question regarding multi variate functions

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When we have a simple one variable function $y = f(x)$, $f(x)$ gives the height of function/curve at an arbitrary point $x$, but if we have a multivariate function $u = f(x,y)$, what does it give us?

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Why not visiting Wolfram Alpha and playing around with different functions $u(x,y)$?

Example 1Example 2 Example 3