What are Linear functions?
A linear function is a function that can be represented as a straight line on the coordinate plane. The exponent of the variable is always equal to 1 . A linear function is represented by \(f(x) = mx + b\), where \(m\) and \(b\) are real numbers. When the linear function is represented on the graph as a line, then it is written as, \(y = mx + b\), where m is the slope of the line, \(b\) is the y-intercept of the line, \(x\) is the independent variable, and \(y\) or \(f(x)\) is the dependent variable.
Example of a linear equation: \(f(x) = 7x – 8\), where \(f(x)\) is the function representing the dependent variable in the equation, 7 is the slope, \(x\) is the independent variable and 8 is the intercept of the line.
What are Nonlinear functions?
A nonlinear function is a function whose plotted graph does not form a straight line but a curved line. The exponent of the variable in a nonlinear equation is greater than 1. A nonlinear function can look like the following examples; \(f(x) = x^2\) which is a quadratic function, \(f(x) = 2^x\) which is an exponential function, and \(f(x) = x^3 – 3x\) which is a cubic function.
Example of a nonlinear function: \(f(x) = 200 – x^2\) where \(f(x)\) is the function and dependent on the equation, 200 is a constant and \(x\) is an independent variable with exponent 2.
Comparing Linear and Nonlinear functions
- A linear function forms a straight line on a graph, while a nonlinear function forms not a straight but a curved line on a graph.
- The linear equation is of the form, \(f(x) = ax + b\), whereas the nonlinear equation can be written in any form other than \(f(x) = ax + b\). For example, \(f(x) = 4x + 8\) is a linear function, whereas \(f(x) = 150 + x^3\) is a nonlinear function.
- Linear functions have a constant slope for any two points on the line, whereas the slope of nonlinear functions is not constant.
- The rate of change for linear functions is always constant whereas, for nonlinear functions, it is variable.
We can observe the differences graphically as well.
We have two functions, one linear function and another nonlinear function.
\(f(x) = 200 – 14x\) (linear function)
\(f(x) = 300 – 15x^2\) (nonlinear function)
Now, let us obtain the value of the function \(f(x)\) for the below set of values of \(x\).
The table for linear functions will be:
If you notice from the table, \(x\) increases by 1 and \(y\) decreases by 14 from one set to the next. Hence, the rate of change is constant.
A similar set of values of for nonlinear functions will be:
If you notice from the table, as \(x\) increases by 1, the value of \(y\) changes by different amounts from one set to next. Hence, the rate of change is not constant.
How to identify functions from tables?
Here are two tables with the values of \(x\) and \(f(x)\).
In the first table, as the value of x increases by 5, the value of \(f(x)\) increases by 10. The change is constant for both sets of values. Thus, this is an example of a linear function.
In the second table, as \(x\) increases, \(f(x)\) increases but not at a constant rate. The rate of change is not constant. Thus, this is an example of a nonlinear function.
Solved Examples
Example 1. Have a look at these two graphs and state which one is linear and which one is nonlinear.
Solution: In the above two graphs, the first graph shows a straight line whereas the second graph shows a curved line. As we have learned, a linear function is graphed as a straight line and nonlinear functions as curved lines.
Thus, the first graph is of a linear function and the second graph is of a nonlinear function.
Example 2. Which of the following functions are nonlinear?
a) \(f(x) = 7\)
b) \(f(x) = \cos x\)
c) \(f(x) = 2^x – 6\)
Solution: Let us have a look at the functions:
a) \(f(x) = 7\) can be written as \(f(x) = 0x + 7\), which is in the form of \(f(x) = mx + c\) and it’s a linear function.
b) \(f(x) = \cos x\) is a trigonometric function and thus it’s a nonlinear function.
c) \(f(x) = 2^x – 6\) is an exponential function, and hence it’s a nonlinear function.
Thus, (a) is a linear function and (b) and (c) are nonlinear functions.
Example 3. Does each equation represent a linear or nonlinear function? Explain.
a) \(y = 2(x – 4)\)
b) \(y = \frac{3}{x}\)
c) \(y = x^2 – 4\)
Solution: Let us have a look at the equations.
a) In this equation, \(y = 2(x – 4)\) can be written as \(y = 2x – 8,\) which is in slope-intercept form of \(y = mx + b\). The function has a constant rate of change. Thus, it’s a linear function.
b) In this equation, \(y = \frac{3}{x}\) cannot be rewritten in slope-intercept form. The function doesn’t have a constant rate of change. Hence, it’s a nonlinear function.
c) In this equation, \(y = x^2 – 4\) is in an exponential form and it cannot be written in slope-intercept form. This is a nonlinear function.
Example 4. The following table shows the bank balances of Joe and Mitchell for the last 5 years. Graph the data and check if there has been any consistent growth for both of them.
Solution: We will plot the points for both Joe and Mitchell.
As you can see, Joe’s points appear on a straight line, while Mitchell’s points are on a curved line. Both the graphs are positive. Now, Joe’s graph has a constant growth with a constant rate of change of $100, while Mitchell’s graph doesn’t have a constant growth but has a curve. This shows that Joe had a constant growth rate in the last 5 years.
Example 5. Which of the following graphs represents a nonlinear function?
Solution: Let us observe the graphs. As we have learned, nonlinear functions don’t represent straight lines but curved lines on graphs. Hence, from the above graphs, graphs a, b, and c are nonlinear functions and d is a linear function.
The exponential value of a variable in a function determines whether its graph will be a straight or a curved line. If the exponent is equal to 1 then the graph is a straight line and such functions are called linear functions. However, if the exponent of the variable in an equation is more than 1 then the graph of the equation is not a straight line. Such equations fall under nonlinear functions.
There are different types of nonlinear functions such as exponential functions, trigonometric functions, quadratic functions, polynomials, and logarithmic functions.