Linear vs. Nonlinear Functions - Definition & Examples

Linear and Non-Linear Functions

Before we talk about linear vs. nonlinear functions, let's quickly review just what a function is.

In math, think of a function like a little machine. We give it an input. The job of the function is to produce an output for each input we give it.

If the input is like a penny into a gumball machine, then the output is like the gumball. The function is the gumball machine itself - the thing changing the input to the output.

Here's an example of a simple function:

Linear Functions

Linear functions are functions that appear as straight lines when they are graphed.

Because of this, linear functions follow the standard equation of a line, which is called "slope-intercept form". Even if an equation doesn't look like it's in slope-intercept form when you first see it, you can almost always transform it to get it there, using the Addition and Subtraction Property of Equality and the Multiplication and Division Property of Equality.

Function $B$ and Function $C$ are linear functions, because they fit into that point-slope form template (think of $C$ as having $+0$ for $b$). No matter what you do you won't be able to get Function $A$ to fit into that form!

Nonlinear Functions

Nonlinear functions are those that are NOT straight lines when they are graphed.

Because of this, nonlinear functions have many forms. They do not follow the equation of a line.

There are many standard non-linear functions, such as quadratic equations.

Linear vs. Nonlinear: Visual examples

Here is an example of a linear function.

This function can be described as "linear" because its graph is a straight line. Notice that you don't see any exponents, which means that the highest possible exponent of a variable (known as the order or degree) is $1$.

Next, here is an example of a nonlinear function.

Notice highest exponent of a variable is $2$, which is greater than $1$. This means that the graph of $g(x)$ curves, which means that it isn't a straight line (it's "non-linear").

(Video) How to Distinguish Between Linear & Nonlinear : Math Teacher Tips

by eHowEducation

This video explains the difference between linear and non-linear functions.

$\large{\text{Summary}}$

Linear function: function that is a straight line and has degree $1$.

Non-linear function: function that is not a straight line and has a degree other than $1$.

There are a couple examples in the video and a couple more examples below.

Here are some examples of linear functions!

Now, here are some examples of $\color{Red}{\text{non-linear functions}}$.