A nonlinear function is most easily described as whatever is not a linear function. Learn what a linear function is and be able to define a nonlinear function; using examples, learn how to distinguish between the two.

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## What are Linear Functions?

The easiest way to define a nonlinear function is as a function that is not a linear function. Therefore, in order to understand what a nonlinear function is, it is essential to understand what a **linear function** is. Graphically, a linear function is a function whose graph is a line. Algebraically, a linear function can be defined as a polynomial with the highest exponent equal to 1 or a horizontal line (*y* = *c* where *c* is a constant).

Linear functions can also be described as a function with a constant slope (rate of change of *y* with respect to *x*). In other words, the slope of the line between any two points of the function is the same.

For example, *y* = 2*x* + 3 is a linear function. Notice it is a polynomial with the highest exponent equal to 1. Also, if we consider some random points that satisfy the equation, say (-1, 1), (0, 3), and (7, 17), we see that the slope of the line between any two pairs of these is the same.

- (-1, 1) and (0, 3): Slope: (3 – 1) / (0 – (-1)) = 2 / 1 = 2
- (0, 3) and (7, 17): Slope: (17 – 3) / (7 – 0) = 14 / 7 = 2
- (-1, 1) and (7, 17): Slope: (17 – 1) / (7 – (-1)) = 16 / 8 = 2

The slope of the line between any two of these points is 2, and this is true for any two points that satisfy the equation *y* = 2*x* + 3. Thus, the slope of the function is constant. The graph of *y* = 2*x* + 3 is shown below, and we see that the graph is a graph of a line.

## Nonlinear Functions

Now that we understand what a linear function is, let’s define a **nonlinear function**. As we stated earlier, nonlinear functions are functions that are not linear functions. Therefore, they have the opposite properties of a linear function.

The graph of a linear function is a line. Thus, the graph of a nonlinear function is not a line. Linear functions have a constant slope, so nonlinear functions have a slope that varies between points. Algebraically, linear functions are polynomials with the highest exponent equal to 1 or of the form *y* = *c* where *c* is constant. Nonlinear functions are all other functions.

An example of a nonlinear function is *y* = *x*^2. This is nonlinear because, although it is a polynomial, its highest exponent is 2, not 1. Also, if we consider some random points that satisfy the equation, say (-3, 9), (-1, 1), and (4, 16), we see that when we calculate the slope of the line between these points, we get different results.

(-3, 9) and (-1, 1): Slope: (1 – 9) / (-1 – (-3)) = -8 / 2 = -4

(-3, 9) and (4, 16): Slope: ((16 – 9) / (4 – (-3)) = 7 / 7 = 1

The slope of the line between different points that satisfy the function is different for different points considered, so the slope varies. Lastly, when we look at the graph of *y* = *x*^2, shown below, it is obvious that this is not the graph of line. Therefore, we see that this is not a linear function. It is a nonlinear function.

## Linear or Nonlinear

Based on all this information, if we want to determine if a function is a nonlinear function, we can do this in a few different ways.

### Solved Examples

Example1: Solve the Linear equation 9(x + 1) = 2(3x + 8)

Solution:

9(x + 1) = 2(3x + 8)

Expand each side

9x + 9 = 6x + 16

Subtract 6x from both the sides

9x + 9 – 6x = 6x + 16 – 6x

3x + 9 = 16

Substract 9 from both the sides

3x + 9 – 9 = 16 – 9

3x = 7

Divide each by 3

3x/3 = 7/3

x= 7/3

Example 2 : Solve the nonlinear equation

3×2 – 5x + 2 = 0

Solution:

3×2 – 5x + 2 = 0

Factorising

3×2 – 3x – 2x + 2 = 0

3x(x – 1) – 2(x – 1) = 0

( 3x – 2)( x – 1) = 0

(3x – 2) = 0 or (x – 1) = 0

x = 2/3 or x = 1

### Quiz Time

Q. Solve the following linear equation and find the value of x

- 3(5x + 6) = 3x – 2
- (2x +9)/5 = 5

Q. Solve the nonlinear equations

- 7×2 = 8 – 10x
- 3(x2 – 4) = 5x

## FAQs

**What is a nonlinear function example?**

Nonlinear Function – A function whose graph is not a line or part of a line. Example: – As **you inflate a balloon, its volume increases**.

**How do you know if a function is nonlinear?**

Simplify the equation as closely as possible to the form of **y = mx + b**. Check to see if your equation has exponents. If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.

**How do you write a nonlinear equation?**

The general form of a nonlinear equation is **ax2 + by2 = c**, where a, b, c are constants and a0 and x and y are variables.