A function that represents a line that is straight on the coordinate plane is referred to as a linear function. For instance, the equation y = 3x – 2 depicts a straight line on a coordinate plane, and as such, it depicts a linear function. Because f(x) can stand in for y, this function can be expressed in mathematical terms as f(x) = 3x – 2.

The definition of a linear function, as well as its graph, domain, and range, will be covered in detail in the following paragraphs of this article. In addition to this, we will discover how to recognise a linear function and how to locate its inverse.

Linear function 

A linear function takes the form f(x) = mx + b, where m and b are both real numbers. This form describes a straight line. Do you not think it resembles the slope-intercept form of a line, which is represented by the equation y = mx + b? The answer to your question is yes, and the reason for this is that the graph of a linear function is a line. Here,

The value of’m’ indicates the slope of the line, while ‘b’ indicates where the line meets the y-axis.

‘x’ is the independent variable

The value of y, or f(x), is known as the dependent variable.

Equation of a Linear Function and Some Examples

The function f(x) = x, which represents a line that runs through the origin, is known as the parent linear function. An equation for a linear function looks like this: f(x) = mx + b. Here are some examples of linear functions.

f(x) = 3x – 2 

f(x) = -5x – 0.5

f(x) = 3

Example of a Linear Function Taken from Real Life

The linear function has a variety of uses in real life, some of which are listed below.

• A movie streaming service typically charges a membership fee of $4.50 per month in addition to charging an additional fee of $0.35 for each movie download. 
• Now, the total cost of the monthly subscription is represented by the linear function f(x) = 0.35x + 4.50, where x is the number of movies that are downloaded in a given month.
• For printing logos on t-shirts, a t-shirt company will charge a one-time fee of $50 in addition to the standard shirt price of $7. Therefore, the overall cost can be calculated using the linear function f(x) = 7x + 50, where x is the number of t-shirts being purchased.
• In linear programming problems, an objective function is typically represented by a linear function. This is done to help minimise losses or maximise profits, depending on the nature of the problem.

Figuring Out What a Linear Function Is

If information about a function is presented as a graph, then the function is considered linear if the graph is in the form of a line. If the information about the function is presented in algebraic form, then we can determine whether or not it is linear by examining whether or not it has the form f(x) = mx + b. However, in order to determine whether the data presented in the form of a table represents a linear function:

• Calculate the varying x-values that have been found.
• Determine the y-values that differ from one another.
• Investigate whether or not there is a consistent pattern in the ratio of the difference in y-values to the difference in x-values.

Constructing a Graph of a Linear Function

We are aware that in order to graph a line, all we need are any two points that are on it. If we can locate two points, we can easily connect them using a line and then extend it along both sides of the triangle. The equation f(x) = mx + b represents the graph of a linear function.

an increasing line when m > 0

a decreasing line when m < 0

a horizontal line when m = 0

A graph of a linear function can be constructed in one of two ways.

• Through the process of locating two points on it.
• Through the utilisation of its slope and y-intercept.

Conclusion

A linear function takes the form f(x) = mx + b, where m and b are both real numbers. This form describes a straight line. The value of’m’ indicates the slope of the line, while ‘b’ indicates where the line meets the y-axis.’x’ is the independent variable

The value of y, or f(x), is known as the dependent variable. A movie streaming service typically charges a membership fee of $4.50 per month in addition to charging an additional fee of $0.35 for each movie download. If information about a function is presented as a graph, then the function is considered linear if the graph is in the form of a line. If the information about the function is presented in algebraic form, then we can determine whether or not it is linear by examining whether or not it has the form f(x) = mx + b.