There are two approaches to linearity. One, more abstract, and another, traditionally characterized as a function of the first degree on an argument. We will briefly mention the former, but go into more details for the latter.

Abstract concept of linearity implies that a function

Example of a such a function is a real function

Indeed,

Here and now we will be talking about more "down to earth" concept of linear functions. Linear function is usually considered on a domain of all real numbers and taking values (having range) in a co-domain of the same set of all real numbers. In some cases it can be considered as a function of a complex argument with a complex range, but it's rare. We will only consider real linear functions.

The characteristic property of linear functions is their expression as an equality of a form

where

Let's mention that in case

The first observation we'd like to make is that this function can be applied to any real number. Therefore, the domain of this function is a set of ALL real numbers.

How about its range? Does it cover all real numbers? The answer is yes. Here is a proof.

For instance, we would like to know if some number

= (R − B) + B = R

Therefore, if an argument equals to

Our second point is that a linear function defines a one-one correspondence between a set of all real numbers as arguments to this function and the same set of all real numbers as its range. Obviously, any real number as an argument is converted by a linear function into unique real number as a value of this function. But, to establish a one-to-one correspondence, this is not enough. We have to prove that different arguments are transformed into different values. Let's prove it.

Assume that two different arguments,

If from two equal values we subtract the same value

If two equal values are divided by the same value

We have assumed in the beginning that

Our next point is related to a monotonic character of a linear function. More precisely, if a linear function

Let's prove this monotonic character of a linear function.

First, consider the case when

Smaller number, if multiplied by a positive number

Smaller number, if added to any number

This proves that for a positive coefficient

Consider now a case of negative coefficient

Again, assume we have two values of argument,

Smaller number, if multiplied by a negative number

Larger number, if added to any number

This proves that for a negative coefficient

Why do we avoid coefficient

The function

In addition, there is a well defined class of polynomial functions expressed as polynomials of its argument in a form

This class is classified by the highest degree the argument should be raised to, in this case it's