Linear Function
A linear function is a function that represents a straight line on the coordinate plane. For example, y = 3x  2 represents a straight line on a coordinate plane and hence it represents a linear function. Since y can be replaced with f(x), this function can be written as f(x) = 3x  2.
In this article, we are going to learn the definition of a linear function along with its graph, domain, and range. We will also learn how to identify a linear function and how to find its inverse.
What is a Linear Function?
A linear function is of the form f(x) = mx + b where 'm' and 'b' are real numbers. Isn't it looking like the slopeintercept form of a line which is expressed as y = mx + b? Yes, this is because a linear function represents a line, i.e., its graph is a line. Here,
 'm' is the slope of the line
 'b' is the yintercept of the line
 'x' is the independent variable
 'y' (or f(x)) is the dependent variable
A linear function is an algebraic function. This is because it involves only algebraic operations.
Linear Function Equation
The parent linear function is f(x) = x, which is a line passing through the origin. In general, a linear function equation is f(x) = mx + b and here are some examples.
 f(x) = 3x  2
 f(x) = 5x  0.5
 f(x) = 3
Real Life Example of Linear Function
Here are some reallife applications of the linear function.
 A movie streaming service charges a monthly fee of $4.50 and an additional fee of $0.35 for every movie downloaded. Now, the total monthly fee is represented by the linear function f(x) = 0.35x + 4.50, where x is the number of movies downloaded in a month.
 A tshirt company charges a onetime fee of $50 and $7 per Tshirt to print logos on Tshirts. So, the total fee is expressed by the linear function f(x) = 7x + 50, where x is the number of tshirts.
 The linear function is used to represent an objective function in linear programming problems, to help minimize the close, or maximize the profits.
How to Find a Linear Function?
We use the slopeintercept form or the pointslope form to find a linear function. The process of finding a linear function is the same as the process of finding the equation of a line and is explained with an example.
Example: Find the linear function that has two points (1, 15) and (2, 27) on it.
Solution:
The given points are (x_{1}, y_{1}) = (1, 15) and (x₂, y₂) = (2, 27).
Step 1: Find the slope of the function using the slope formula:
m = (y₂  y_{1}) / (x₂  x_{1}) = (27  15) / (2  (1)) = 12/3 = 4.
Step 2: Find the equation of linear function using the point slope form.
y  y_{1} = m (x  x_{1})
y  15 = 4 (x  (1))
y  15 = 4 (x + 1)
y  15 = 4x + 4
y = 4x + 19
Therefore, the equation of the linear function is, f(x) = 4x + 19.
Identifying a Linear Function
If the information about a function is given as a graph, then it is linear if the graph is a line. If the information about the function is given in the algebraic form, then it is linear if it is of the form f(x) = mx + b. But to see whether the given data in a table format represents a linear function:
 Compute the differences in xvalues.
 Compute the differences in yvalues
 Check whether the ratio of the difference in yvalues to the difference in xvalues is always constant.
Example: Determine whether the following data from the following table represents a linear function.
x  y 

3  15 
5  23 
7  31 
11  47 
13  55 
Solution:
We will compute the differences in xvalues, differences in y values, and the ratio (difference in y)/(difference in x) every time and see whether this ratio is a constant.
Since all numbers in the last column are equal to a constant, the data in the given table represents a linear function.
Graphing a Linear Function
We know that to graph a line, we just need any two points on it. If we find two points, then we can just join them by a line and extend it on both sides. The graph of a linear function f(x) = mx + b is
 an increasing line when m > 0
 a decreasing line when m < 0
 a horizontal line when m = 0
There are two ways to graph a linear function.
 By finding two points on it.
 By using its slope and yintercept.
Graphing a Linear Function by Finding Two Points
To find any two points on a linear function (line) f(x) = mx + b, we just assume some random values for 'x' and substitute these values in the function to find the corresponding values for y. The process is explained with an example where we are going to graph the function f(x) = 3x + 5.
 Step 1: Find two points on the line by taking some random values.
We will assume that x = 1 and x = 0.  Step 2: Substitute each of these values in the function to find the corresponding yvalues.
Here is the table of the linear function y = 3x + 5.x y 1 3(1)+5 = 2 0 3(0)+5 = 5  Step 3: Plot the points on the graph and join them by a line. Also, extend the line on both sides.
Graphing a Linear Function Using Slope and yIntercept
To graph a linear function, f(x) = mx + b, we can use its slope 'm' and the yintercept 'b'. The process is explained again by graphing the same linear function f(x) = 3x + 5. Its slope is, m = 3 and its yintercept is (0, b) = (0, 5).
 Step 1: Plot the yintercept (0, b).
Here, we plot the point (0, 5).  Step 2: Write the slope as the fraction rise/run and identify the "rise" and the "run".
Here, the slope = 3 = 3/1 = rise/run.
So rise = 3 and run = 1.  Step 3: Rise the yintercept vertically by "rise" and then run horizontally by "run". This results in a new point.
(Note that if "rise" is positive, we go up and if "rise" is negative, we go down. Also, if "run" is positive", we go right and if "run" is negative, we go left.)
Here, we go up by 3 units from the yintercept and thereby go right by 1 unit.  Step 4: Join the points from Step 1 and Step 2 by a line and extend the line on both sides.
Domain and Range of Linear Function
The domain of a linear function is the set of all real numbers, and the range of a linear function is also the set of all real numbers. The following figure shows f(x) = 2x + 3 and g(x) = 4 −x plotted on the same axes.
Note that both functions take on real values for all values of x, which means that the domain of each function is the set of all real numbers (R). Look along the xaxis to confirm this. For every value of x, we have a point on the graph.
Also, the output for each function ranges continuously from negative infinity to positive infinity, which means that the range of either function is also R. This can be confirmed by looking along the yaxis, which clearly shows that there is a point on each graph for every yvalue. Thus, when the slope m ≠ 0,
 The domain of a linear function = R
 The range of a linear function = R
Note:
(i) The domain and range of a linear function is R as long as the problem has not mentioned any specific domain or range.
(ii) When the slope, m = 0, then the linear function f(x) = b is a horizontal line and in this case, the domain = R and the range = {b}.
Inverse of a Linear Function
The inverse of a linear function f(x) = ax + b is represented by a function f^{1}(x) such that f(f^{1}(x)) = f^{1}(f(x)) = x. The process to find the inverse of a linear function is explained through an example where we are going to find the inverse of a function f(x) = 3x + 5.
 Step 1: Write y instead of f(x).
Then the above equation becomes y = 3x + 5.  Step 2: Interchange the variables x and y.
Then we get x = 3y + 5.  Step 3: Solve the above equation for y.
x  5 = 3y
y = (x  5)/3  Step 4: Replace y by f^{1}(x) and it is the inverse function of f(x).
f^{1}(x) = (x  5)/3
Note that f(x) and f^{1}(x) are always symmetric with respect to the line y = x. Let us plot the linear function f(x) = 3x + 5 and its inverse f^{1}(x) = (x  5)/3 and see whether they are symmetric about y = x. Also, when (x, y) lies on f(x), then (y, x) lies on f^{1}(x). For example, in the following graph, (1, 2) lies on f(x) whereas (2, 1) lies on f^{1}(x).
Piecewise Linear Function
Sometimes the linear function may not be defined uniformly throughout its domain. It may be defined in two or more ways as its domain is split into two or more parts. In such cases, it is called a piecewise linear function. Here is an example.
Example: Plot the graph of the following piecewise linear function.
\(f(x)=\left\{\begin{array}{ll}
x+2, & x \in[2,1) \\
2 x3, & x \in[1,2]
\end{array}\right.\)
Solution:
This piecewise function is linear in both the indicated parts of its domain. Let us find the endpoints of the line in each case.
When x ∈ [2, 1):
x  y 

2  2 + 2 = 0 
1 (hole in this case as 1 ∉ [2, 1) ) 
1 + 2 = 3 
When x ∈ [1, 2]:
x  y 

1  2(1)  3 = 1 
2  2(2)  3 = 1 
The corresponding graph is shown below:
Important Notes on Linear Functions:
 A linear function is of the form f(x) = mx + b and hence its graph is a line.
 A linear function f(x) = mx + b is a horizontal line when its slope is 0 and in this case, it is known as a constant function.
 The domain and range of a linear function f(x) = ax + b is R (all real numbers) whereas the range of a constant function f(x) = b is {b}.
 These linear functions are useful to represent the objective function in linear programming.
 A constant function has no inverse as it is NOT a oneone function.
 Two linear functions are parallel if their slopes are equal.
 Two linear functions are perpendicular if the product of their slopes is 1.
 A vertical line is NOT a linear function as it fails the vertical line test.
☛ Related Topics:
Examples on Linear Functions

Example 1: The relationship between Celsius degrees and Fahrenheit degrees is linear. Some equivalent values are shown in the table below. Find the linear function representing the given data.
Celsius (°C) Fahrenheit (°F) 5 41 10 50 15 59 20 68 Solution:
To find the linear function, it is sufficient to consider any two points from the table.
Let (x_{1}, y_{1}) = (5, 41) and (x₂, y₂) = (10, 50).
The slope, m = (y₂  y_{1}) / (x₂  x_{1}) = (50  41) / (10  5) = 9/5.
Using the pointslope form,
y  y_{1} = m (x  x_{1})
y  41 = (9/5) (x  5)
y  41 = (9/5) x  9
y = (9/5) x + 32
From the table, the independent variable is C and the dependent variable is F. So the linear relationship is, F = (9/5) C + 32.
Answer: The linear relationship between Celsius and Fahrenheit is, F = (9/5) C + 32.

Example 2: The cost (in dollars) of renting a car is represented by C(x) = 30 x + 20, where x is the number of days the car is rented for. Then what is the cost of renting the car for 10 days?
Solution:
To find the cost of renting the car for 10 days, substitute x = 10 in the given linear function.
C(10) = 30(10) + 20 = 300 + 20 = 320
Answer: The cost of renting the car for 10 days = $320.

Example 3: Considering the scenario of Example 2, if Ryan paid the total rent to be $470, then for how many days did he rent the car?
Solution:
The given linear function is C(x) = 470, where 'x' is the number of days that car is rented for.
470 = 30x + 20
Solving the above linear equation,
450 = 30x
x = 450/30 = 15
Answer: Ryan rented the car for 15 days.
FAQs on Linear Function
What is the Definition of a Linear Function?
A linear function is a function whose graph is a line. Thus, it is of the form f(x) = mx + b where 'm' and 'b' are real numbers. Here, 'm' is the slope and 'b' is the yintercept of the linear function.
What is the Formula to Find a Linear Function?
Since a linear function represents a line, all formulas used to find the equation of a line can be used to find the equation of a linear function. Thus, the linear function formulas are:
 Standard form: ax + by + c = 0
 Slopeintercept form: y = mx + b
 Pointslope form: y  y_{1} = m (x  x_{1})
 Intercept form: x/a + y/b = 1
Note that y can be replaced with f(x) in all these formulas.
What is Linear Function Table?
Sometimes, the data representing a linear function is given in the form of a table with two columns where the first column gives the data of the independent variable and the second column gives the corresponding data of the dependent variable. This is called the linear function table.
What is the Difference Between a Linear Function and a Nonlinear Function?
The graph of a linear function is a line whereas the graph of a nonlinear function is NOT a line. The equation of a linear function is of the form f(x) = ax + b (i.e., it is a linear expression), whereas the equation of a nonlinear function can be of any other forms than ax + b.
How to Graph a Linear Function?
To graph a linear function, find any two points on it by assuming some random numbers either for the dependent or for the independent variable and find the corresponding values of the other variable. Just plot those two points and join them by a straight line by extending the line on both sides.
What is the Domain and Range of a Linear Function?
The domain and range of a linear function f(x) = ax + b where a ≠ 0 is the set of all real numbers. If a = 0, the domain is still the set of all real numbers but the range is the set {b}. Sometimes, the domain and range in a problem may be restricted to some interval.
What is a Linear Function Equation?
The linear function equation is the slopeintercept form. Thus, it is expressed as f(x) = mx + b where m is the slope and b is the yintercept of the line.
What are Linear Function Examples?
f(x) = 2x + 3, f(x) = (1/5) x  7 are some examples of linear function. For real life examples of a linear function, click here.
How to Determine a Linear Function?
We can determine a linear function in the following ways.
 If the equation of a function is given, then it is linear if it is of the form f(x) = ax + b.
 If the graph of a function is given, then it is linear if it represents a line.
 If a table of values representing a function is given, then it is linear if the ratio of the difference in yvalues to the difference in xvalues is always a constant.
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