Linear Function

A linear function is a mathematical function that creates a straight line when graphed. It can be described by the formula:

y = mx + b

In Algebra, a linear function represents astraight line in the 2-D or 3-D cartesian plane. This is why such functions are called "linear." They are functions that involve variables and constants, but no exponent values greater than one.

The general form of a linear function is expressed as:

y = f(x) = mx + c

where,

• y is the Dependent Variable,
• x is the Independent Variable,
• m is the Coefficient of x,
• c is the Constant Value.

Key Characteristics:

• A function whose graph is a straight line is called a linear function.
• It is a polynomial function of degree one (1) because the highest power of the variable x is 1.
• The function relates the dependent variable (y or f(x)) to the independent variable (x) through a linear relationship.

Examples of Linear Function

Various example of the linear function are,

• y = f(x) = 2x + 1
• y = f(x) = -3x - 2
• y = f(x) = 5

Non-Linear Function

A non-linear function is the function that are not linear in nature, i.e. the graph of these function do not represent the straight line. The graph of these functions represents, circle, parabola, hyperbola, etc. These function are called,

• Quadratic Functions,
• Parabolic Functions,
• Hyperbolic Function, etc.

Linear Function Formula

Linear Function Formula is used to represent the objective function of the linear programming problems, which helps to maximize profits or minimize input cost. The data is provided in a LPP is a linear function. In general a linear function, on is in the form, f(x) = ax + b and the purpose of the Linear Programming Problems is to maximize or minimize the linear function under some conditions that are given in the LPP.

How to Find Linear Function

A linear function connecting at least two coordinates is easily found using the point slope form or intercept form of a line. As a linear function is the equation of straight line. It is found using equation of line concept. This is explained in the example added below,

Example: Find the Linear function when two points on the function are, (-1, 2) and (3, 4)

Solution:

Given Points,

• (x₁, y₁) = (-1, 2)
• (x₂, y₂) = (3, 4)

Slope of Line(m) = (y₂ - y₁)/(x₂ - x₁)

m = (4 - 2)/(3 - {-1}) = 2/4 = 1/2

Now the linear function is,

y - y₁ = m(x - x₁)
y - 2 = 1/2(x - {-1})
y - 2 = 1/2(x + 1)
2y - 4 = x + 1
x - 2y + 5 = 0

This is the required linear function.

Graphing of a Linear Function

We know that graph of linear equation represents the straight line and to draw a straight line we need at least two point and joining those two points and stretching the line in both the direction gives the required straight line.

The graph of a linear function f(x) = mx + b is shown in the image added below as,

• Case 1: When m > 0

The image added below shows the linear function when m > 0,

• Case 2: When m < 0

The image added below shows the linear function when m < 0,

• Case 3: When m = 0

The image added below shows the linear function when m = 0,

Graphing a Linear Function by Finding Two Points

To discover two pinpoints on a linear function (line) f(x) = mx + b, consider some unexpected values for 'x' and have to replace these values to find the connected values of y. 

This method is presented by an instance where we are proceeding to graph the function f(x) = 2x + 4.

Step 1: Find two points on the line by first taking two random value of x

• Let the values be: x = 0 and x = 1

Step2: Find the value of the y with the respective value of the x.

So, the two points on the line are (0, 4) and (1, 6).

Step 3: Plot the point on the graph and join them to get the graph of required linear function.

Graphing of Linear Function Using Slope and Y-intercept

To graph a linear function using slope and y-intercept form, we first the linear function in the standard slope as,

f(x) = mx + b

where, m is slope of line and the y intercept is b.
For example, f(x) = 2x + 4

• slope of line = 2
• y-intercept = 4
• point on y-axis = (0, 4)

Now to find plot the line we follow the steps added below,

Step 1: Firstly Plot the y-intercept (0, b) i.e. (0, 4)

Step 2: Now the slope in fraction is represented as rise/run

Here,

slope = 2 = 2/1 = rise/run

So, rise = 2 and run = 1

Step 3: Rise the y-intercept vertically by "rise" and then run horizontally by "run". This results in a new point.

Here, we move 2 units vertically in the direction of y-axis and move horizontally 1 unit in direction of x-axis.

Step 4: Now join the points from Step 1 and Step 3 we get the required graph of linear function.

Domain and Range of Linear Function

Domain of the linear function is the collection of all real numbers, and the range of a linear function is the collection of all numbers that are found by substituting the value of in the linear function.

The general form of the linear function is y = ax + b and if a ≠ 0 then the domain and range of the function is,

• Domain of Linear Function = R
• Range of Linear Function = R

Note: When the slope, m = 0, then the linear function f(x) = b is a horizontal line, and in this case,

• Domain = R
• Range = {b}

Inverse of Linear Function

Inverse of the linear function f(x) = ax + b is represented as by a function f^-1(x) such that,

• f(f^-1(x)) = f^-1(f(x)) = x

Inverse of the function is explained using the example added below,

Example: Find the inverse of f(x) = 2x + 4

Solution:

Step 1: Write the given linear function as,

y = 2x + 4

Step 2: Now interchange the variables x and y

x = 2y + 4

Step 3: Solve the above equation to get y

x - 4 = 2y
y = (x - 4)/2

Step 4: Replace y by f^-1(x) and it is the inverse function of f(x).

f^-1(x) = (x - 4)/2

Note: f(x) and f^-1(x)are always symmetric with respect to the line y = x

Piecewise Linear Function

A function that is linear on some domain of the function or the function that is function in a specific interval of domain is called the Piecewise Linear Function. Example of Piecewise Linear Function is,

• f(x) = 2x, x ∈ [-5, 4)
• f(x) = -x + 11, x ∈ [4, 12]

Read More

• Relation and Function
• Function
• Types of Functions

Solved Examples on Linear Functions

Example 1: Find the linear function that has two points (-2, 17) and (1, 26) on it.

Solution:

Given Points,

• (x₁, y₁) = (-2, 17)
• (x₂, y₂) = (1, 26)

Step1: Firstly find the slope of the function using the slope formula:

m = (y₂ - y₁) / (x₂ - x₁)
m = (26 - 17) / (1 - (-2))
m = 9/3 = 3

Step2: Now find the equation of linear function using the point-slope form 

y - y₁ = m (x - x₁)
y - 17 = 3 (x - (-2))
y - 17 = 4 (x + 2)
y - 17 = 4x + 8
y = 4x + 25

So, the equation of linear function is, f(x) = 4x + 25

Example 2: Check whether the data set represents a linear function or not.

Solution:

Plot the points in the table

X	Y	(Difference in X)/(Difference in Y)
3 ⇣+2 5	16 ⇣+8 23	⇒ 8/2 = 4
5 ⇣+2 7	23 ⇣+8 31	⇒ 8/2 = 4
7 ⇣+4 11	31 ⇣+16 47	⇒ 16/4 = 4
11 ⇣+2 13	47 ⇣+8 55	⇒ 8/2 = 4

As all the numbers in the last column are equal, the given table represents the linear function.

Example 3: Plot Linear Function Graph y = 3x + 2

Solution:

Take some value of x and find its corresponding y-values.

x	y = 3x + 2
1	3 × 1 + 2 = 5
2	3 × 2 + 2 = 8
3	3 × 3 + 2 = 11

Example 4: Plot the graph of the following equation 3x + 2y − 4 = 0

Solution:

Given Linear Function, 3x + 2y - 4 = 0

3x + 2y = 4
3x/4 + 2y/4 = 1
x/(4/3) + y/(2) = 1

Comparing with x/a + y/b = 1

• a = 4/3
• b = 2

Now, point on x-axis is (a, 0) = (4/3, 0)
Point on y-axis is (0, b) = (0, 2)

Plotting these points on the graph and joining them we get the required linear function.

Practice Questions on Linear Function

Question 1: Plot the graph of the following equation 2x + y − 8 = 0.

Question 2: Plot the graph of the following equation x + y − 1 = 0.

Question 3: Find the linear function that has two points (1, 3) and (-2, 4) on it.

Question 4: Find the linear function that has two points (-1, -2) and (1, 2) on it.