A nonlinear function is most easily described as whatever is not a linear function. Learn what a linear function is and be able to define a nonlinear function; using examples, learn how to distinguish between the two.
What are Linear Functions?
The easiest way to define a nonlinear function is as a function that is not a linear function. Therefore, in order to understand what a nonlinear function is, it is essential to understand what a linear function is. Graphically, a linear function is a function whose graph is a line. Algebraically, a linear function can be defined as a polynomial with the highest exponent equal to 1 or a horizontal line (y = c where c is a constant).
Linear functions can also be described as a function with a constant slope (rate of change of y with respect to x). In other words, the slope of the line between any two points of the function is the same.
For example, y = 2x + 3 is a linear function. Notice it is a polynomial with the highest exponent equal to 1. Also, if we consider some random points that satisfy the equation, say (-1, 1), (0, 3), and (7, 17), we see that the slope of the line between any two pairs of these is the same.
- (-1, 1) and (0, 3): Slope: (3 – 1) / (0 – (-1)) = 2 / 1 = 2
- (0, 3) and (7, 17): Slope: (17 – 3) / (7 – 0) = 14 / 7 = 2
- (-1, 1) and (7, 17): Slope: (17 – 1) / (7 – (-1)) = 16 / 8 = 2
The slope of the line between any two of these points is 2, and this is true for any two points that satisfy the equation y = 2x + 3. Thus, the slope of the function is constant. The graph of y = 2x + 3 is shown below, and we see that the graph is a graph of a line.
Now that we understand what a linear function is, let’s define a nonlinear function. As we stated earlier, nonlinear functions are functions that are not linear functions. Therefore, they have the opposite properties of a linear function.
The graph of a linear function is a line. Thus, the graph of a nonlinear function is not a line. Linear functions have a constant slope, so nonlinear functions have a slope that varies between points. Algebraically, linear functions are polynomials with the highest exponent equal to 1 or of the form y = c where c is constant. Nonlinear functions are all other functions.
An example of a nonlinear function is y = x^2. This is nonlinear because, although it is a polynomial, its highest exponent is 2, not 1. Also, if we consider some random points that satisfy the equation, say (-3, 9), (-1, 1), and (4, 16), we see that when we calculate the slope of the line between these points, we get different results.
(-3, 9) and (-1, 1): Slope: (1 – 9) / (-1 – (-3)) = -8 / 2 = -4
(-3, 9) and (4, 16): Slope: ((16 – 9) / (4 – (-3)) = 7 / 7 = 1
The slope of the line between different points that satisfy the function is different for different points considered, so the slope varies. Lastly, when we look at the graph of y = x^2, shown below, it is obvious that this is not the graph of line. Therefore, we see that this is not a linear function. It is a nonlinear function.
Linear or Nonlinear
Based on all this information, if we want to determine if a function is a nonlinear function, we can do this in a few different ways.
Example1: Solve the Linear equation 9(x + 1) = 2(3x + 8)
9(x + 1) = 2(3x + 8)
Expand each side
9x + 9 = 6x + 16
Subtract 6x from both the sides
9x + 9 – 6x = 6x + 16 – 6x
3x + 9 = 16
Substract 9 from both the sides
3x + 9 – 9 = 16 – 9
3x = 7
Divide each by 3
3x/3 = 7/3
Example 2 : Solve the nonlinear equation
3×2 – 5x + 2 = 0
3×2 – 5x + 2 = 0
3×2 – 3x – 2x + 2 = 0
3x(x – 1) – 2(x – 1) = 0
( 3x – 2)( x – 1) = 0
(3x – 2) = 0 or (x – 1) = 0
x = 2/3 or x = 1
Q. Solve the following linear equation and find the value of x
- 3(5x + 6) = 3x – 2
- (2x +9)/5 = 5
Q. Solve the nonlinear equations
- 7×2 = 8 – 10x
- 3(x2 – 4) = 5x
Nonlinear Function – A function whose graph is not a line or part of a line. Example: – As you inflate a balloon, its volume increases.
Simplify the equation as closely as possible to the form of y = mx + b. Check to see if your equation has exponents. If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.
The general form of a nonlinear equation is ax2 + by2 = c, where a, b, c are constants and a0 and x and y are variables.