Graph of the Functions of Two or More Variables

What Are Multivariable Functions?

Functions of two or more variables involve multiple independent inputs. $$ f:R^n \rightarrow R $$

The output can be a single real value (R), an ordered pair (R²), or an m-tuple of real values (R^m).

$$ f:R^n \rightarrow R^m $$

Illustrative Examples
The Graph of a Function of Two Variables

Illustrative Examples

Example 1

Consider a function of two real variables:

$$ f:R^2 \rightarrow R $$

One such example is:

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

Note. The independent variables can also be written as components of a vector, with $ x_1 $ and $ x_2 $ representing the first and second components respectively: $$ f(x_1,x_2) = x_1^2 - x_2 + x_1 x_2 $$

Below is the graph of the function in three-dimensional space, with axes x, y, and f(x, y):

3D surface of the function f(x, y) = x² - y + xy

Example 2

Another function of two real variables:

$$ f:R^2 \rightarrow R $$

is defined as:

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

Here is its graph in the three-dimensional space (x, y, f(x, y)):

3D surface of the function f(x, y) = x² - y²

Example 3

Now consider a function of three real variables:

$$ f:R^3 \rightarrow R $$

defined as:

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

Note. A function may be defined over multiple variables without necessarily using all of them in its expression. For instance, the function below is still a function of three variables, even though only x and y appear explicitly: $$ f(x,y,z) = x^2 + y^2 $$

In this case, the graph cannot be visualized, as it resides in four-dimensional space: x, y, z, and f(x, y, z).

The Graph of a Function of Two Variables

The graph of a function $ f:A \rightarrow \mathbb{R} $, where $ A \subseteq \mathbb{R}^2 $, is the set of all points $ (x, y, z) \in \mathbb{R}^3 $ such that $ (x, y) $ lies in the domain $ D_f $, and $ z = f(x, y) $.

$$ \text{graph} = \{ (x, y, z) \in \mathbb{R}^3 \mid (x, y) \in D_f, \ z = f(x, y) \} $$

Here, $ \mathbb{R}^3 $ is the Cartesian product $ \mathbb{R} \times \mathbb{R} \times \mathbb{R} $ of the variables x, y, and z.

Note. For a function of a single variable, the graph is a curve in the plane: $$ \text{graph} = \{ (x, y) \in \mathbb{R}^2 \mid x \in D_f, \ y = f(x) \} $$ For a function of two variables, $ z = f(x, y) $, the graph becomes a surface in three-dimensional space: $$ \text{graph} = \{ (x, y, z) \in \mathbb{R}^3 \mid (x, y) \in D_f, \ z = f(x, y) \} $$

In general, the graph of a function of $ n $ variables is a subset of $ \mathbb{R}^{n+1} $ and is defined as follows:

$$ \text{graph} = \{ (\vec{x}, y) \in \mathbb{R}^{n+1} \mid \vec{x} \in D_f, \ y = f(\vec{x}) \} $$

Here, $ \vec{x} $ is a vector with $ n $ components, and $ y $ is a real number.

$$ \vec{x} = (x_1, x_2, \ldots, x_n) $$

Note. This notation closely mirrors the one-variable case, $ y = f(x) $, where $ x $ is a real number: $$ \text{graph} = \{ (x, y) \in \mathbb{R}^2 \mid x \in D_f, \ y = f(x) \} $$ In a multivariable context, however, $ x $ becomes a vector in $ \mathbb{R}^n $: $$ \text{graph} = \{ (\vec{x}, y) \in \mathbb{R}^{n+1} \mid \vec{x} \in D_f, \ y = f(\vec{x}) \} $$

And so on.