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) = x2 is nonlinear as it is a quadratic function.
- f(x) = 2x is nonlinear as it is an exponential function.
- f(x) = x3 - 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
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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) = x2 - 2x + 2, f(x) = ln x, f(x) = ex, 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 non-linear 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.
