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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.
Non Linear Functions | Mathematics for SAT

Let's graph the table now.
Non Linear Functions | Mathematics for SAT

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.
Non Linear Functions | Mathematics for SAT

Let us determine whether this table denotes a nonlinear function by using the steps mentioned above.
Non Linear Functions | Mathematics for SAT

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.
Non Linear Functions | Mathematics for SAT

Linear and Nonlinear Functions

Here are the differences between linear and nonlinear functions.
Non Linear Functions | Mathematics for SAT

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.

Nonlinear Function Examples

Example: Does the following table represents a nonlinear function?
Non Linear Functions | Mathematics for SATSol: Yes.
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.

The document Non Linear Functions | Mathematics for SAT is a part of the SAT Course Mathematics for SAT.
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