A multivariable function takes more than one input. For example,

$$f(x,y)=x^2+y^2$$

assigns one real number to each allowed point $(x,y)$.

More generally, a scalar-valued function of $n$ variables can be written

$$f:\mathbb R^n\to\mathbb R.$$

Domain

The domain is the set of input points for which the function is defined. For

$$f(x,y)=\sqrt{1-x^2-y^2},$$

we need

$$x^2+y^2\le1,$$

so the domain is the closed unit disk.

Graphs

For a function $z=f(x,y)$, the graph consists of points

$$(x,y,f(x,y))$$

in three-dimensional space. For example,

$$z=x^2+y^2$$

forms an upward-opening paraboloid.

Functions of three or more variables cannot usually be visualized through an ordinary graph in physical space, so other descriptions become important.

Level sets

A level set contains the input points that give the same output value:

$$f(x,y)=c.$$

For $f(x,y)=x^2+y^2$, the level curves satisfy

$$x^2+y^2=c,$$

so positive levels are circles centered at the origin.

Level curves and level surfaces reveal the geometry of a scalar field without requiring its full graph.