Multivariable functions can be built from familiar single-variable functions by combining several inputs inside the same expression.

For example,

$$f(x,y)=e^{-(x^2+y^2)}$$

combines a polynomial expression with the exponential function.

Domains from component restrictions

Every part of the expression contributes its own domain restrictions. For

$$g(x,y)=\ln(4-x^2-y^2),$$

the logarithm requires

$$4-x^2-y^2>0,$$

so

$$x^2+y^2<4.$$

The domain is the open disk of radius $2$.

Symmetry

The expression itself often reveals geometric symmetry. A function depending only on

$$x^2+y^2$$

has the same value at every point the same distance from the origin. Its level curves are therefore circles.

Similarly, a function depending on $x^2+y^2+z^2$ has spherical level surfaces.

Common shapes

Simple polynomial functions already produce important surfaces. For example,

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

is a paraboloid, while

$$z=x^2-y^2$$

is a saddle surface.

Recognizing the elementary functions inside a multivariable expression helps predict its domain, symmetry, level sets and broad geometric behavior before calculus is applied.