Instead of having just one input \(x\) determining the output \(y\) (like \(y = f(x)\)), functions of multiple variables have two or more inputs that together determine the output.
Think of a topographical map: the height (output) depends on your location specified by two coordinates, latitude and longitude (inputs). Or the temperature in a room depends on your \((x, y, z)\) position.
A function of \(n\) variables\(f\) assigns a unique output value \(z\) (usually a real number) to each input vector \((x_1, x_2, ..., x_n)\) from a domain \(D\) in \(n\)-dimensional space (\(\mathbb{R}^n\)).
Notation: \(z = f(x_1, x_2, ..., x_n)\) or \(z = f(\mathbf{x})\), where \(\mathbf{x}\) is the input vector.
Domain: The set of all possible input vectors \((x_1, ..., x_n)\) for which the function is defined.
Range: The set of all possible output values \(z\).
Two Variables (\(z = f(x, y)\)): The graph is a surface in 3D space. For each point \((x, y)\) in the domain (on the xy-plane), the height of the surface above/below that point is \(z = f(x, y)\).
Three or More Variables: Direct graphing is impossible (requires >3 dimensions). We often visualize using:
Level Sets (Contour Maps): For \(z = f(x, y)\), level sets are curves in the xy-plane where \(f(x, y) = c\) (constant). Like contour lines on a topographical map showing points of equal height.
Level Surfaces: For \(w = f(x, y, z)\), level surfaces are surfaces in 3D space where \(f(x, y, z) = c\).
Loss Functions: Often functions of many variables (the model's parameters/weights \(w_1, w_2, ..., w_n\)). The loss \(L\) depends on all weights: \(L = f(w_1, ..., w_n)\). Optimization aims to find the weights that minimize this function.
Prediction Models: A model predicting house price (\(y\)) based on size (\(x_1\)) and location (\(x_2\)) is a function \(y = f(x_1, x_2)\).
Image Representation: Pixel intensity can be a function of coordinates \(I = f(\text{row}, \text{column})\).
The concepts of limits and continuity extend to multiple variables, but become more complex. A limit requires the function to approach the same value regardless of the path taken towards the point \((x_0, y_0, ...)\). Continuity requires the limit to exist, the function to be defined, and the limit to equal the function value, just like the single-variable case.