A homogeneous function is a type of mathematical function that exhibits a specific scaling property. If a function f(x) is homogeneous of degree n , it means that when all its inputs are multiplied by a constant k , the output is scaled by k^n . In simpler terms, f(kx) = k^n f(x) . This property is often used in various fields, including economics and physics, to describe relationships that maintain their form under proportional changes. Homogeneous functions can be classified into different degrees, such as linear (degree 1) or quadratic (degree 2). They are particularly useful in optimization problems and in the study of Cobb-Douglas functions in economics, where they help analyze production and utility functions. Understanding homogeneous functions aids in simplifying complex equations and modeling real-world scenarios where proportionality is key.