Linear Function

A linear function is a polynomial function of degree 1, meaning the highest power of x is 1. It represents a constant rate of change determined by the slope m. The graph of a linear function is always a straight line.

A linear function is commonly written in the slope–intercept form:

It relates the independent variable x with the dependent variable y (or f(x)).

Various examples 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 a function whose graph does not form a straight line on the Cartesian plane. In such functions, the variables may have powers greater than one or appear in different forms, such as quadratic, exponential, or trigonometric expressions.

Examples:

• y = x² (Quadratic function)
• y = x³ (Cubic function)
• y = 2ˣ (Exponential function)

Linear Function Graph

Steps to Graph a Linear Function

• Find two points that satisfy the equation y = mx + b.
• Plot these points on the Cartesian (X–Y) plane.
• Draw a straight line passing through the plotted points

Finding a Linear Function

A linear function can be found using the slope formula and the point-slope form of a line when two points on the line are known.

Example: Find the linear function that passes through the points (1, 3) and (4, 9).

Solution:

Given points:
(x₁, y₁) = (1, 3)
(x₂, y₂) = (4, 9)

Step 1: Find the slope

m = (y₂ − y₁) / (x₂ − x₁)
m = (9 − 3) / (4 − 1)
m = 6 / 3
m = 2

Step 2: Use the point-slope form

y − y₁ = m(x − x₁)

y − 3 = 2(x − 1)

Step 3: Simplify the equation

y − 3 = 2x − 2
y = 2x + 1

Therefore, the linear function is:
f(x) = 2x + 1

Graphing of a Linear Function

To graph a linear function, we need at least two points that satisfy the equation. Plot these points on the coordinate plane and connect them to form the required line.

For a linear function f(x) = mx + b:

• The graph is increasing when m > 0.
• The graph is decreasing when m < 0.
• The graph is horizontal when m = 0.

There are two ways to graph a linear function:

• By finding two points on the line.
• By using the slope and the y-intercept.

Graphing a Linear Function by Finding Two Points

To graph a linear function f(x) = mx + b, choose any two values of x and substitute them in the function to find the corresponding values of y.

Example: Graph the function f(x) = 2x + 4

Step 1: Choose two values of x

Let x = 0 and x = 1.

Step 2: Find the corresponding values of y

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

Step 3: Plot the points (0, 4) and (1, 6) on the coordinate plane. Join them with a straight line and extend the line in both directions to obtain the graph of the linear function.

Graphing of Linear Function Using Slope and Y-intercept

A linear function can be graphed using the slope–intercept form:

f(x) = mx + b, where m is the slope and b is the y-intercept.

Example: Graph the function f(x) = 2x + 4

Slope = 2
y-intercept = 4
Point on y-axis = (0, 4)

Step 1: Plot the y-intercept (0, 4) on the coordinate plane.

Step 2: Express the slope as a fraction.

Slope = 2 = 2/1
Rise = 2, Run = 1

Step 3: From the point (0, 4), move 2 units up and 1 unit to the right to get another point (1, 6).

Step 4: Draw a straight line passing through (0, 4) and (1, 6) and extend it in both directions.

Domain and Range of a Linear Function

The domain of a linear function is the set of all possible values of x, and the range is the set of all possible values of y obtained from the function.

For a linear function f(x) = mx + b (where m ≠ 0):

• Domain = ℝ (all real numbers)
• Range = ℝ (all real numbers)

This means the function is defined for every real value of x, and the graph extends infinitely in both directions.

Note: When the slope m = 0, the function becomes f(x) = b, which represents a horizontal line.

In this case:

• Domain = ℝ
• Range = {b}

The figure above shows the graphs of f(x) = 2x + 3 and g(x) = 4 − x on the same coordinate plane, indicating that the domain and range of linear functions are all real numbers (ℝ).

Inverse of Linear Function

The inverse of a linear function f(x) = ax + b is a function f⁻¹(x) such that:

• f(f⁻¹(x)) = f⁻¹(f(x)) = x

Example: Find the inverse of the function f(x) = 3x + 5.

Solution:

Step 1: Write the function as
y = 3x + 5

Step 2: Interchange the variables x and y
x = 3y + 5

Step 3: Solve the equation for y
x − 5 = 3y
y = (x − 5) / 3

Step 4: Replace y with f⁻¹(x)

f⁻¹(x) = (x − 5) / 3

Note: The graphs of f(x) and its inverse f⁻¹(x) are symmetric about the line y = x.

Piecewise Linear Function

A piecewise linear function is a function that is defined by different linear expressions over different intervals of its domain. Each part of the function represents a straight line within its specified interval.

Example:

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

In this case, the function follows one linear rule for the first interval and another linear rule for the second interval.

Also Check

• Relation and Function
• Function
• Types of Functions

Solved Problems

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

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

Step 1: Find the slope

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

Step 2: Use the point-slope form

y − y₁ = m(x − x₁)

y − 17 = 3(x − (-2))
y − 17 = 3(x + 2)

Step 3: Simplify the equation

y − 17 = 3x + 6
y = 3x + 23

Therefore, the linear function is

f(x) = 3x + 23

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

Plot the points in the table

X	Y	(Difference in Y)/(Difference in X)
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

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: + 2y − 4 = 0

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 Problems

Problem 1: Plot the graph of the following equation: -2x + y − 8 = 0, and also identify two points that lie on this equation.

Problem 2: Plot the graph of the following equation: x + y − 1 = 0, and also identify two points that lie on this equation.

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

Answer:

1. (2, 4), (4, 0)
2. (-2, 3), (2, -1)
3. 3y+x-10=0