Functions of Multiple Variables

In many cases, functions can depend on more than one or even more than two input variables. These are called functions of multiple variables, where the inputs are independent, but they collectively determine a single output. Let’s explore this concept in detail.

Definition: Functions of Two Variables

A function \( f \) that depends on two variables \( x \) and \( y \) is written as:

\[ z = f(x, y), \]

where:

\( x \) and \( y \) are the independent variables.
\( z \) is the output (dependent variable).
\( x \) takes values from a set \( X \), and \( y \) takes values from a set \( Y \).
The domain of the function is the cartesian product \( X \times Y \), i.e., all possible ordered pairs \( (x, y) \).

Thus, the function is described as:

\[ f : X \times Y \to Z, \quad \text{where } z = f(x, y). \]

Example 1: Paraboloid Surface

Let \( f : \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be defined by:

\[ f(x, y) = x^2 + y^2. \]

The function computes the sum of the squares of \( x \) and \( y \).
The graph of this function represents a paraboloid surface in 3-dimensional space, where \( z \) is the height above the \( xy \)-plane.

Example 2: Plane Surface

Let \( f(x, y) = 3x + 4y \).

This function calculates a weighted sum of \( x \) and \( y \).
The graph of this function represents a plane in 3-dimensional space.

Extending to Three Variables: Functions of \( x, y, z \)

A function can also depend on three variables, such as:

\[ w = f(x, y, z), \]

where:

\( x, y, z \) are the independent variables.
\( w \) is the output (dependent variable).
The domain of the function is the cartesian product \( X \times Y \times Z \), which consists of all ordered triplets \( (x, y, z) \).

Example 3: Spherical Distance Function

Let \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \).

This function calculates the distance from the origin to the point \( (x, y, z) \) in 3D space.
The output \( w \) represents the radial distance from the origin.

Example 4: Weighted Sum

Let \( f(x, y, z) = 2x + 3y + 4z \).

This function adds weighted contributions of \( x, y, \) and \( z \).
It is a hyperplane in 4-dimensional space, which is hard to visualize geometrically but is mathematically important.

Generalization to More Variables

Functions can be extended to n variables, written as:

\[ f(x_1, x_2, \dots, x_n), \]

where \( x_1, x_2, \dots, x_n \) are independent inputs, and the domain is the cartesian product \( X_1 \times X_2 \times \dots \times X_n \).

Example

Consider the function:

\[ f(x, y) = 3x^2 + 4y^2, \]

where \( x \) and \( y \) are independent variables, and \( f(x, y) \) represents the output.

Evaluation of \( f(x, y) \) at Specific Points

Point 1: \( (x, y) = (1, 2) \)

To evaluate the function at \( (x, y) = (1, 2) \), substitute \( x = 1 \) and \( y = 2 \) into the expression for \( f(x, y) \):

\[ f(1, 2) = 3(1)^2 + 4(2)^2. \]

Simplify:

\[ f(1, 2) = 3(1) + 4(4) = 3 + 16 = 19. \]

Point 2: \( (x, y) = (-1, -3) \)

To evaluate the function at \( (x, y) = (-1, -3) \), substitute \( x = -1 \) and \( y = -3 \) into the expression for \( f(x, y) \):

\[ f(-1, -3) = 3(-1)^2 + 4(-3)^2. \]

Simplify:

\[ f(-1, -3) = 3(1) + 4(9) = 3 + 36 = 39. \]

The values of \( f(x, y) \) are:

\[ f(1, 2) = 19, \quad f(-1, -3) = 39. \]

This example illustrates how to evaluate a function of two variables by substituting specific values of \( x \) and \( y \) into the given formula.