# Non-linear Function in Discrete mathematics

On the basis of the name of a non-linear function, it is a function that is NOT linear. The non-linear function can be described as a function that does not create a line on the graph. The graph of a non-linear function will be anything but not a line. In this section, we will learn about the definition, graph, and examples of the non-linear function.

### What is a non-linear Function?

A non-linear function can be described as a function that does not draw a straight line on the graph. The graph of a non-linear function can be a curve, but it must not be a line.

For example: If we have a pound that has 100 fish in the starting and every week the fishes become double, in this case, we can model this situation with the help of function f(x) like f(x) = 100 (2)x, where f(x) is used to indicate the number of fishes, and x is used to indicate the number of weeks. Now we can make a table and graph of this function, which is described as follows:

x y
0 100
1 200
2 400
3 800

Now we will graph the above table, which is described as follows:

In the above graph, we can see that there is NO straight line. Hence this graph has a non-linear function. In the above graph of a non-function, there is no uniform way in the slope of this graph. We can show a non-linear function with the help of a table of values, an equation, or a graph. Now we will learn each of them. There are some examples of non-linear functions, i.e., polynomial functions, cubic functions, quadratic functions, etc.

### Table of Non-linear function

There are some steps that are used to find whether a table of values finds a linear function or not. These steps are described as follows:

Step 1: In this step, we will determine the difference between every two consecutive x values.

Step 2: In this step, we will determine the difference between every two consecutive y values.

Step 3: In this step, we will determine the corresponding ratios of differences of y and differences of x.

Step 4: The function will be linear only if the ratios calculated in step 3 are not the same.

To understand this, we will consider a table of values, which is described as follows:

x y
3 15
5 23
9 33
11 41
13 43

Now we will use the above steps and find out whether the above table shows a non-linear function.

The function is non-linear because there is not the same number of ratios between the differences of y to the differences of x.

### Equation of Non-linear function

A linear function is used to contain the form f(x) = ax+b. Since a linear function can be described as a function that will not be linear. The non-linear equation can contain any type of form, but that form must not be f(x) = ax+b. There are various examples of non-linear functions, and some of them are described as follows:

• f(x) = x2 is a non-linear and this function is also a quadratic function.
• f(x) = 2x is a non-linear function and this function is also a exponential function.
• f(x) = x3 - 3x is a non-linear function and this function is also a cubic function.

### Graphs of Non-linear Function

If there is a function that is not linear, then it will be known as the non-linear function. A non-linear function will be represented by those types of functions that do not generate a straight line in the graph. In the following graphs, we can see that there is no straight line in all the images. All the graphs show the non-linear function.

### Linear and Non-linear Functions

The linear and non-linear functions contain a lot of differences, which are described as follows:

Linear Functions Non-linear functions
If we use a graph to indicate the linear function, then that graph will be in a straight line. If we use a graph to indicate the non-linear function, then that graph will NOT be a line.
The linear function must be contained the form f(x) = ax+b. The non-linear function can contain any type of form, but that form must not be f(x) = ax+b.
The slope of linear function for any two points on the curve must be a constant. The slope of non-linear function on a graph for any two points must NOT be the same.
In case of a linear function, if we draw a table, then the ratio between differences of y and differences of x will be a constant. In the case of a non-linear function, if we draw a table, then the ratio between differences of y and differences of x will NOT be a constant.

### Important points of Non-linear function

• If there is a graph of a function that is not a line, then it will be non-linear.
• If there is an equation that does not contain the form f(x) = ax+b, then that function will be known as the non-linear function.
• With the help of an objective function z = ax+by, we can either represent the linear or non-linear function.
• There are some examples of non-linear functions such as exponential functions, polynomial functions, logarithm functions, etc.

### Examples of Non-linear

The no-linear is used to contain the various examples, and some of them are described as follows:

Example 1: In this example, there are three functions, and we have to determine which function is non-linear. These functions are described as follows:

1. f(x) = 4
2. f(x) = 3x - 7
3. f(x) = sin x

Solution:

1. The first function is f(x) = 4 and we can write it in the form f(x) = 0x+4. This function is same as the form f(x) = ax+b. Hence, we can say that this function is a linear function.
2. The second function is f(x) = 3x - 7 and this function contains the exponent function. Hence we can say that this function is a non-linear function.
3. The third function is f(x) = sin x and this function contains the trigonometric function. Hence we can say that this function is a non-linear function.

Hence, only (2) and (3) both are non-linear functions.

Example 2: In this example, we have a table with some values, and we have to show whether this table shows a linear function or not. The table is described as follows:

x y
1 5000
2 2500
3 1250
4 625
5 312.5

Solution:

As we can see, 1, 1, 1, and 1 are the differences between every two successive values of x.

Similarly, -2500, -1250, -625, and -312.5 are the differences between every two successive values of y.

The ratio between the x values and y values are -2500, -1250, -625, and -312.5, and these values are not the same.

Therefore, we can say that the function which is shown in the above table is non-linear.

Hence, this table is non-linear.

Example 3: In this example, there are four graphs, and we have to show whether they show non-linear or not.

Solution: According to the definition of a non-linear function, the graph of a non-linear function must not be a line. Graph numbers (a), (b), and (c) do not contain a straight line, whereas graph (d) contains a straight line. So we can say that graphs (a), (b), and (c) are non-linear graphs, but graph (d) is not a non-linear graph or a linear graph.