A thorough understanding of linear functions is important for students aiming to take various competitive exams in the United States, including the SAT, ACT, GRE, and AP Calculus, in which algebraic concepts, graphing, and functions are highly tested. Linear functions, expressing straight-line relations between variables, are the backbone of most mathematical concepts that students will encounter on these exams. 

• For example, in the SAT Math section, students are often required to analyze linear equations, find slopes, and graphs. 
• Understanding linear functions develops problem-solving abilities and enhances performance in algebra and geometry sections. 

This article gives a comprehensive description of linear functions, their formulas, graphing methods, real-world applications, and solved examples to improve understanding and enhance exam preparation.

What is a Linear Function?

As already discussed a linear function is a function which when plotted on a graph forms a straight line. Also, it is a polynomial function with degree utmost 1 or 0.

A linear function in two variables has one dependent variable and one independent variable.

We can represent a linear function as y =f(x)= mx + b.

Here, x is the independent variable and y is the dependent variable.

Also, ‘m’ is the constant term and can also be called as the y-intercept of the function.

When x = 0, then, b acts as the coefficient of the independent variable and is known as the slope. It also gives the rate of change of dependent variables.

As a linear function involves algebraic operations, it is an algebraic function.

Linear Function Graph

In order to graph a linear function or a line, we need only two points lying on the line. To form a line, we just need to connect two points and then extend it in both directions. The graph of a linear function f(x) = mx + b is:

• An increasing line when m > 0.
• A decreasing line when m < 0.
• And a horizontal line when m = 0.

We can draw a graph for a linear function in two ways:

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

Plotting a Graph of Linear Function by Finding Two Points

To plot a graph of linear function y =mx + b by finding two points on it, we assume some value of x and then substitute this value in the equation to get the corresponding value of y.

Let us understand this using an example:

Example: Consider the function y = 3x + 5.

Step 1: Firstly, find two points on the line by taking some random values of x. Let us assume x = -1 and x = 0.

Step 2: Then, substitute these values in the equation of the linear function to get the corresponding value of y.

For x = -1, y = 3(-1) + 5 = -3 + 5 = 2

For x = 0, y = 3(0) + 5 = 0 + 5 = 5

So, the coordinates to be plotted become: (-1, 2) and (0, 5).

Step 3: Plot the two points on the graph and join them using a lime. Also, we can extend the line in both directions.

Learn about Greatest Integer Function

Plotting a Graph of Linear Function by Using Slope and y-intercept

To plot a graph of the linear function y = mx + b, we need to use slope ‘m’ and y-intercept ‘b’. Let us understand this concept using the same function, i.e. y = 3x +5.

Here, slope m = 3, and y-intercept (0, b) = (0, 5).

Step 1: We need to plot the y-intercept, i.e. (0,5) on the graph.

Step 2: Write the slope in the form of a fraction, like rise/run.

Here, slope = 3 = 3/1 = rise/run

Here rise = 3, and run = 1

Step 3: Rise the y-intercept in the vertical direction by ‘rise’ and then run in the horizontal direction by ‘run’, we get a new point.

Here, we can go vertically above by 3 points and then horizontally by 1 point.

So, the new point becomes (1, 8).

Step 4: Join the two points from Step 1, and Step 3, and then extend it in both directions to get the desired line.

Learn about General Equation of a Line

Linear Function Table

Take a look at the table below to check the notation of the ordered pair in normal as well as function form:

By examining the values of x and y in the table we can verify the linear function. It is to be noted that for a linear function the rate of change of y with respect to x remains constant and is called the slope of the line.

Let us consider the given table:

We can see from the table above that the rate of change of x with respect y is 3. As a linear function this can be written as y = x +3.

Learn about Difference Between Relation and Function

Linear Function Formula

Expression of linear function to plot a straight line graph is termed as the formula of a linear function. We can express a linear equation as y = mx + b.

Here m is the slope of the straight line and b is the y-intercept and (x, y)is the coordinate. This is known as the slope formula. However, when it comes to expressing as functions, it can be done as:

f(x) = Ax + B, where x is the independent variable.

Learn about linear equations in one variables and linear equations in two variables

Domain and Range of Linear Function

The domain and range of all linear functions is a set of all real numbers. Let us understand using the following graph:

Let the linear function be: f(x) = 2x +3, and g(x) = 4 -x. Plot these two linear functions on the same graph.

From the graph, we can see that both the linear functions take all the real values of x. This means that the domain of each of the above given functions is set of all real numbers.

Also, we can see that the output of each of the above linear functions ranges from negative infinity to positive infinity. That means that the range of the functions is also a set of all real numbers.

This is possible for the linear functions where slope is not equal to zero. For the functions with slope equal to zero, f(x) = b, the domain of the function is a set of all real numbers but the range of the function is {b}.

Inverse of Linear Functions

We can represent the inverse of a linear function f(x) = ax + b by \(f^{-1}x\) such that \(f(f^{-1}(x))=f^{-1}(f(x))=x\).

Let us understand the process of finding the inverse of a linear function with the help of an example:

Example: Consider the function f(x) = 3x +5.

Step 1: Replace f(x) by y in the above linear function. The equation becomes y = 3x +5.

Step 2: Interchange the variables x and y. The equation now becomes x = 3y + 5.

Step 3: Now, solve the equation obtained in step 2 to find the value of y.

3y + 5 = x

3y = x – 5

y = (x – 5)/3

Step 4: Replace y with \(f^{-1}x\) and we get the inverse of the given linear function.

\(f^{-1}x\) = (x-5)/3

It is to be noted that both the functions f(x) and \(f^{-1}x\) are symmetric to a function y =x.

Let us plot the two functions y = 3x + 5 and y = (x -5)/3 on a graph to see whether they are symmetric to y = x.

Piecewise Linear Functions

There may be some linear functions that are not defined uniformly throughout its domain. The function might be defined in two or more ways as its domain is split in two or more ways. Such a function is called a piecewise linear function.

Let us consider an example to understand a piecewise linear function:

Example: Consider the function f(x) = x + 2 for \(x \in [-2, 1)\) and f(x) = 2x -3 for \(x \in [1, 2]\)

The function is linear in both the given parts of the domain. Let us find the endpoints of a line in each case.

When \(x \in [-2, 1)\)

Consider x = -2 , then, y = -2 + 2 = 0

And x = 1, then, y = 1 + 2 = 3

When \(x \in [1, 2]\)

Consider x = 1, then, y = 2(1) -3 = 2 – 3 = -1

And x = 2, then, y = 2(2) – 3 = 4 – 3 = 1

The graph is shown below:

Real Life Examples of Linear Functions

Some of the real life examples of linear functions are listed below:

• The monthly charges of a movie streaming company are $4.50 with an additional fee of $0.30 for every movie that is downloaded. We can represent the total monthly fee with the help of a linear function as:

f(x) = 0.30x + 4.50. Here x represents the number of movies downloaded within a month.

• One-time fee for a T-shirt company is $7.50 with an additional fee of $0.65 for printing a logo. So, we can express the total fee as a linear function as:

f(x) = 7.50x + 0.65. Here, x represents the number of T-shirts.

Properties of Linear Function

Let us check how linear functions can be used to plot two points on the graph:

• Relation: This is the group of ordered pairs.
• Variables: This is a symbol that represents a quantity in a mathematical expression.
• Linear Function: An expression where each term is either constant or a product of a constant with a variable of degree 1.
• Function: This is a relation between a set of inputs and a set of permissible outputs. This says each input is related to exactly one output.
• Steepness: This is the rate at which a function deviates from the reference.
• Direction: This can be increasing, decreasing, horizontal or vertical.

Learn about Linear Equations in Two Variables

Linear Function Solved Examples

Que 1: Cost of renting a car is represented by a linear function C(x) = 30x +20. Here x is the number of days for which the car is rented. Find the cost of renting a car for 10 days.

Solution: As the cost of renting a car is given as: C(x) = 30x +20.

We have to find the cost of renting a car for 10 days.

Let us put x = 10 in the given function:

C(10) = 30(10) + 20

C(10) = 300 + 20 = 320.

Therefore, the cost of renting a car is $320.

Que 2: For the above linear function, if James paid an amount of $470. Find the number of days for which the car was rented.

Solution: Given that the amount paid by James = $470.

Also, the given linear function is C(x) = 30x +20

Replacing C(x) by 470, we get:

470 = 30x +20

470 – 20 = 30 x

450 = 30 x

x = 450/30 = 15.

Therefore, the car was rented for 15 days.

Therefore, it is evident that Linear functions are the building blocks of most mathematical topics and are therefore crucial for high performance in US competitive exams such as the SAT, ACT, GRE, and AP Calculus. Knowledge of linear equations, their graphs, and practical applications provides students with problem-solving strategies to approach a wide range of mathematics-based questions. We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.