Nonlinear Function — Definition, Formula & Examples

A nonlinear function is any function whose graph is not a straight line. Its rate of change is not constant, meaning the output does not increase or decrease by the same amount for equal changes in input.

A function $f$ is nonlinear if it cannot be expressed in the form $f(x) = mx + b$ for constants $m$ and $b$ . Equivalently, there exist inputs $x_1$ , $x_2$ , and $x_3$ equally spaced apart such that the differences $f(x_2) - f(x_1)$ and $f(x_3) - f(x_2)$ are not equal.

How It Works

To test whether a function is nonlinear, check if the rate of change between consecutive points stays the same. Pick pairs of input-output values and compute the change in output divided by the change in input. If that ratio ever differs from one pair to the next, the function is nonlinear. Common nonlinear functions include quadratics like

y = x^2

, exponentials like

y = 2^x

, and absolute value functions.

Worked Example

Problem: Determine whether the function given by this table is linear or nonlinear: x = 1, 2, 3, 4 and y = 2, 4, 8, 16.

Step 1: Find the rate of change between the first two points.

\frac{4 - 2}{2 - 1} = \frac{2}{1} = 2

Step 2: Find the rate of change between the next two points.

\frac{8 - 4}{3 - 2} = \frac{4}{1} = 4

Step 3: Compare the rates. Since 2 ≠ 4, the rate of change is not constant.

Answer: The function is nonlinear because the rate of change is not constant. (In fact, this is the exponential function

y = 2^x

Why It Matters

Recognizing nonlinear functions is a key skill in 8th-grade math (standard 8.F.3) and prepares you for studying quadratics, exponentials, and other curves in algebra and beyond. Many real-world relationships — like the area of a circle as its radius grows, or the spread of a population over time — are nonlinear.

Common Mistakes

Mistake: Assuming a function is linear just because its outputs are increasing.

Correction: A function can increase steadily and still be nonlinear. Check whether the rate of change is the same between every pair of consecutive points, not just whether the outputs go up.

Related Terms

Input-Output Table — Used to test if a function is nonlinear
Absolute Value Function — A common example of a nonlinear function
Algebraic Function — Broad category that includes nonlinear functions
Argument of a Function — The input value you plug into any function