Calculus How To

Nonlinear Function

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Types of Functions >

What is a Nonlinear Function?

A nonlinear function is defined as one that isn’t a linear function.

Graphically, a linear function is simply any function that produces a straight line graph. More formally, a straight line produced when the dependent variable (y) changes at a constant rate with the independent variable (y), following the equation y = mx + b. In addition, a linear function has a domain and range of all real numbers. A nonlinear function is a function that doesn’t meet these requirements. They are, in a sense, the opposite of linear functions. In other words, a nonlinear function is any function that:

  • Has a curved, bent or broken graph,
  • Has a domain and range of something other than all real numbers.
  • Has a non-constant rate of change,
  • Fits any equation other than y = mx + b.

The following functions are all nonlinear:

Most polynomial functions are nonlinear functions with one exception: Algebraically, a linear function is a polynomial with a degree (highest exponent) of 1. They are also known as first degree polynomials. Nonlinear functions are everything else (second degree, third degree, …).

Graph of a Linear Function

Linear functions are any functions that produce a straight line graph. So by definition, nonlinear functions produce graphs that aren’t a straight line.

nonlinear function

Linear function (red) and two nonlinear functions: exponential (blue) and polynomial (green). Graph created with

Nonlinear Function vs. Linear Function: Steps

In order to figure out if your function is linear or nonlinear, you have several options. From easiest to hardest, they are:

  • Study the equation. If it neatly fits the equation y = mx + b, then it’s linear.
  • Graph your function. If there is an obvious straight line (with no dips, curves, or changes in direction), then your function is linear. Otherwise, it’s a nonlinear function.
  • Calculate the slope of the line between different points (using “rise over run“). You’re looking for the same rate of change between points. Be careful to graph your function after performing this step though, as some functions (like the absolute value function) might appear to have the same rate of change, except that the absolute value function mirrors itself (one side will have a positive slope and the other a negative slope).

integrable function

An absolute value function has a sharp dip where it changes direction. Therefore, it isn’t linear, but does appear to have the same slope.

Nonlinear Functional Analysis

Nonlinear functional analysis is the study of nonlinear functions. It’s the complement of linear functional analysis. In other words, it’s defined as the study of any function that isn’t linear.


Desmos Graphing Calculator.
The Biology Project. Linear Functions The Basics. Retrieved January 12, 2021 from:

Stephanie Glen. "Nonlinear Function" From Calculus for the rest of us!

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