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Nonlinear Function
A nonlinear function, as its name suggests, is a function that is NOT linear. In other words, the graph of a nonlinear function is NOT a line. i.e., its graph can be anything other than a line.
Let us learn more about nonlinear functions along with its definition, graph, and examples.
What is a Nonlinear Function?
A nonlinear function is a function whose graph is NOT a straight line. Its graph can be any curve other than a straight line. For example, if there are 100 fishes in a pond initially and they become double every week, then this situation can be modeled by the function f(x) = 100 (2)^{x}, where x is the number of weeks and f(x) is the number of fishes. Let us make a table and graph this function making use of the table.
x  y 

0  100 
1  200 
2  400 
3  800 
Let's graph the table now.
The above graph is NOT a line and hence it represents a nonlinear function. From the above graph, we can say that the slope is not uniform on a nonlinear function. A nonlinear function can be described using a table of values, an equation, or a graph. Let us see each of them now. Some of the examples of non linear functions include quadratic functions, cubic functions, polynomial functions.
Nonlinear Function Table
The steps to determine whether a table of values determine a linear function are:
 Find the differences between every two consecutive x values.
 Find the differences between every two consecutive y values.
 Find the corresponding ratios of differences of y and differences of x.
 If all the ratios are NOT same, then only the function is linear.
Consider the following table of values.
x  y 

3  15 
5  23 
9  33 
11  41 
13  43 
Let us determine whether this table denotes a nonlinear function by using the steps mentioned above.
Since all the ratios of differences of y to the differences of x are NOT same, the function is a nonlinear function.
Nonlinear Function Equation
A linear function is of the form f(x) = ax + b. Since a nonlinear function is a function that is not a linear, its equation can be anything that is NOT of the form f(x) = ax+b. Some examples of nonlinear functions are:
 f(x) = x^{2} is nonlinear as it is a quadratic function.
 f(x) = 2^{x} is nonlinear as it is an exponential function.
 f(x) = x^{3}  3x is nonlinear as it is a cubic function.
Non linear Graphs
Since a function that is NOT linear is being called as a nonlinear function, any function whose graph is NOT a straight line should represent a nonlinear function. In the following figure, all graphs represent nonlinear functions as they are NOT straight lines.
Linear and Nonlinear Functions
Here are the differences between linear and nonlinear functions.
Linear Functions  Nonlinear Functions 

A linear function is a function whose graph is a line.  A nonlinear function is a function whose graph is NOT a line. 
Its equation is of the form f(x) = ax + b.  Its equation can be in any form except of the form f(x) = ax + b. 
Its slope is constant for any two points on the curve.  The slope of every two points on the graph is NOT the same. 
In the table of a linear function, the ratio of difference of y and difference of x is a constant.  In the table of a nonlinear function, the ratio of difference of y and difference of x is NOT a constant. 
Tips and Tricks on Nonlinear Functions
 If the graph of a function is NOT a line, its nonlinear.
 If the equation of a function is NOT of the form f(x) = ax + b, then its nonlinear.
 The objective function z = ax + by may be a linear function or a non linear function.
 Rational functions, polynomial functions, exponential functions, logarithmic functions, etc are some examples of nonlinear functions.
☛ Related Topics:
Nonlinear Function Examples

Example 1: Which of the following functions are nonlinear? a) f(x) = 4 (b) f(x) = 3^{x}  7 (c) f(x) = sin x.
Solution:
(a) f(x) = 4 and it can be written as f(x) = 0x + 4 and it is of the form f(x) = ax + b. Hence, it is a linear function.
(b) f(x) = 3^{x}  7 is an exponential function and hence it is nonlinear.
(c) f(x) = sin x is a trigonometric function and hence it is nonlinear.
Answer: Only (b) and (c) are nonlinear functions.

Example 2: Does the following table represents a nonlinear function?
x y 1 5000 2 2500 3 1250 4 625 5 312.5 Solution:
The differences of every two successive values of x are 1, 1, 1, and 1.
The differences of every two successive values of y are 2500, 1250, 625, and 312.5.
Their corresponding ratios are 2500, 1250, 625, and 312.5, which are NOT the same.
Hence the function represented by the table is nonlinear.
Answer: Yes.

Example 3: Which of the following graphs represents non linear functions?
Solution:
We know that a nonlinear function's graph is NOT a line. So among the given graphs, (a), (b), and (c) are nonlinear whereas (d) is a linear function.
Answer: (a), (b), and (c).
FAQs on Nonlinear Function
How do You Know if a Function is Nonlinear?
If a function is NOT linear, then it is known as a nonlinear function. If the equation of a function is NOT of the form f(x) = ax + b then it is a nonlinear function.
Write an Example of Nonlinear Function.
Any function that is not linear is a nonlinear function. So some examples of nonlinear functions are f(x) = x^{2}  2x + 2, f(x) = ln x, f(x) = e^{x}, etc.
What are Different Types of Nonlinear Functions?
A nonlinear function, by its name, is NOT linear. So any function that is not of the form f(x) = ax+b is nonlinear. So different types of nonlinear functions are polynomial functions, exponential functions, logarithmic functions, trigonometric functions, rational functions, modulus function, etc.
How can we Write Nonlinear Function From a Table?
To write a nonlinear function from its table, we need to observe the pattern. Consider the example below.
x  y 

0  3 
1  6 
2  12 
3  24 
Here, we can see that the initial value is 3 and each time it is getting multiplied by 2. So the nonlinear function of this table is f(x) = 3(2)^{x}.
What is the Difference Between Linear and Non Linear Graphs?
A linear function gives a graph that is a straight line. On contrary, a nonlinear function gives a graph that is NOT a line. A part of a nonlinear function may be a graph but the entire graph is NOT made by just one single line.
How to Identify a Nonlinear Function Table?
To identify whether a table represents a nonlinear function, compute the differences of every two rows for both x and y values. Then compute the ratios of corresponding differences (y/x). If all the ratios are NOT the same, then the table represents a nonlinear function.
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