Bivariate Function

A bivariate function (also called a function of two variables) is one that assigns output values when given two input variable values. Graphing of these functions result in a 3D Graph. Such functions are also called binary functions (as its arity is 2).

Bivariate functions are a subset of (the much larger) multivariate functions.

In most cases, we take the domain to be the Real numbers, $\mathbb{R}$. Hence, such functions are called real binary/bivariate functions.

Definition §

Function (Bivariate) ^definition-bivariate

A function $f$ is bivariate if that takes values from a set $X^2$ and transforms/maps them into values in another set $Y$. Alternatively, a bivariate function is a function of two variables. This is notated as:

$$f:D\subseteq X^2\to Y$$

$$y=f(x_1,y_2)$$

Real Bivariate Function ^definition-real-bivariate

If $f$ is a real-valued bivariate function, then it takes values from ${\mathbb{R}}^2$ (the cartesian plane) and maps them to $\mathbb{R}$ (the real numbers):

$$f:D\subseteq {\mathbb{R}}^2\to \mathbb{R}$$

$$z=f(x,y)$$

Graphing §

See Graphing a Bivariate Function

Limits §

See Bivariate Limit