Understanding Linear Functions in Calculus


Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs.

A simple function can be as follows:

[{1, 2}, {2, 4}, {3, 6}, {4, 18}, {5, 10}]

The above function has five pairs where the first members are 1, 2, 3, 4 and 5.

Functions usually have alphabetical letter as their names. So if we term this function ‘f’, which is the most common letter used for functions, then it will be properly written as:

f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8, f(5) = 10

Here are two definitions to keep in mind:

The entire set of first numbers in the function is called a domain and the first members are called arguments. In this particular example, the domain has 5 numbers and the numbers 1, 2, 3, 4 and 5 are the arguments of the function.

The whole set of second numbers in the function is called the range and the second members are called the values. Going back to the above function, the range also has 5 numbers and the numbers 2, 4, 6, 8 and 10 are the values of the function.

As mentioned before, the standard naming of a function is f. Thus we can explain this function in a sentence as follows:

The value of the function (f) at argument 1 is 2, its value at argument 2 is 4, its value at argument 3 is 6, its value at argument 4 is 8 and its value at argument 5 is 10.

Therefore a function can also be defined as a set of assigned values (the second numbers) to arguments (the first numbers)

This can be expanded further to say that the condition is that the first member of every pair is different; therefore each argument of the domain of function ‘f’ gets an exclusive value in its range.

The linear function and its importance to calculus

The linear function is the basic and essential function, on which calculus is based upon. This is a function that has a straight line running through the domain of its graphs.

Such a line can be determined by two points that lie on it. Look at the function [a, f{a}], [b, f{b}]. You can pick an “a” and “b” in the domain and determine this line defined by the two values f{a} and f{b}.

Let’s look at the formula for such a function.

It is possible to determine the linear function for the two values mentioned above by using the following formula.

f{x} = [f{a} x – b/ a – b] + [f{b} x – a/ b – a ]

Effectively this means that the first term is 0 when x is equal to b, and it becomes f{a} when x equals a. The second term is 0 when x is equal to a and it becomes f{b} when x equals b.

Another important aspect of a linear function is its slope.

This is defined as the ratio of the change of function f between x = a and x = b the change in x between the two arguments. The y-intercept is the point at which the line passes the y-axis.

The intercept of the line on the y axis is also an essential part of the linear function.

As we have seen, a linear function can be defined one that has a graph with a straight line, and can be described by its slope and y-intercept.

Special linear functions are often useful and they all have an important and unique property – they all have linear functions whose y-intercepts go through the point 0. Their graphs pass through the origin of the x and y axes. They are aptly called homogenous linear functions, and they all share the same property which is:

Their value at any permutation of two arguments is equal to the same permutations of their values at those arguments.

This can be explained by the following formula:

F{ak + bc} = af{k} + bf{c}

The above property is called the “property of linearity”.

NOTE: not all linear functions have this property of linearity. The property implies that once you know the value of a linear function and any two distinct arguments, then you can find the value at any other point or pair of arguments. This is not always true.

Practical applications of the linear functions

There are several real life applications of calculus linear functions. Remember that this is the most basic function on which other functions are based upon. The function is applied in various fields, such as meteorology, pharmaceuticals, engineering, and a lot more.

Whenever you have to create a graph in a straight line, no matter what the slope or y-intercept is, you are applying this basic principle.

NOTE: One should not confuse linear functions in calculus to linear equations in algebra. They have different properties even if sometimes their graphs can be identical. You can find a graph for a linear equation of algebra having the same slope and y-intercept as a graph for linear function of calculus, but they do not represent the same properties.


Starting off by understanding this basic formula of calculus will make it very easy for you to move on and understand the deeper functions or integration and differentiation. Calculus should not be a behemoth to be feared but a friend to be understood. Try out some basic exercises on the linear functions in calculus and you will get a better grip on the topic.