Question

How do you know if a function is nonlinear?

Answer 1

Hint: The function is nonlinear if its curve is not a line. A nonlinear function has properties opposite to linear functions. A function is said to be a nonlinear function based on three criteria namely, using a graph, using an equation, using a table.

Complete step by step solution:
All the non-linear functions have degrees greater than or equal to two whereas linear ones have degrees equal to one only.
The general representation of the non-linear function is $p{x}^2+q{y}^2=r$
A function is said to be non-linear based on three criteria,
1. Using a graph
2. Using a table
3. Using an equation
Using a graph:
We need to plot the graph for the function given.
If the graph is curved or not straight, then the function is non-linear.
Example1:
$\Rightarrow y=3x+2$
The graph of the function is as follows,

The graph is a straight line.
$\therefore$$y=3x+2$is not a nonlinear function.
Example2:
$\Rightarrow y=x^2+2x+4$
The graph of the function is as follows,

The graph is not a straight line.
$\therefore y=x^2+2x+4$is a nonlinear function.
Using a table:
Let the function be $y=f\left(x\right)$
Draw a table for the values of x and y.
If the differences between the values of y are the same, the function is linear. If the differences are not the same, the function is non-linear.
Example:
$\Rightarrow y=3x^2+2$
The table for the values of x and y is as follows,

x	y
0	2
1	5
2	14

The difference between the values is not the same.
$\therefore y=3x^2+2$is a nonlinear function.
Using an equation:
If the function is not of the form $y=mx+c$, it is said to be a nonlinear function.
Example:
$\Rightarrow y=x^2+x+3$
The above expression cannot be written in the form $y=mx+c$
$\therefore y=x^2+x+3$is a nonlinear function.

Note: In mathematics, Non-linearity is a term used to describe a situation where there is no direct relationship between the independent and the dependent variable. All the functions with a degree greater than or equal to 2 are non-linear.