Non-linear functions are mathematical expressions where the relationship between the independent variable(s) and the dependent variable is not a straight line. Unlike linear functions, which can be represented by a simple equation of the form (y = ax + b), non-linear functions can take various forms, including polynomials, exponentials, logarithms, and more complex combinations.

For example, a cubic function can be expressed as (y = ax^3 + bx^2 + cx + d), where the highest power of (x) is three, indicating a non-linear relationship. Non-linear functions can exhibit a variety of behaviors, such as curves, peaks, and troughs, making them suitable for modeling complex real-world phenomena, such as population growth or economic trends.

In the context of regression analysis, non-linear regression is used when the data shows a curvy trend that linear regression cannot accurately capture. This is particularly important in fields like economics, biology, and engineering, where relationships between variables are often non-linear ¹².