Overview on Algebra of functions

The operations of the functions, specifically the arithmetic operations, are referred to as the algebra of functions. Algebra of functions is primarily concerned with the four arithmetic operations on functions that are listed below:

• Addition of functions
• Subtraction of functions
• Multiplication of functions
• Division of functions

Here are the formulas of all these opeoperation, 

•                   (f+g)(x) = f(x) + g(x) 
•                   (f-g) (x) = f(x) – g(x) 
•                   (f.g) (x) = f(x) . g(x) 
•                   (f/g) (x) = f(x) / g(x) ,  {where g(x) not equal to zero}

There are two other important operations to consider, which are composite functions and inverse functions, in addition to these two.

Algebra of function formulas:

Any arithmetic operation (addition/subtraction/multiplication/division) of two functions is simply the same operation of two independent functions performed on the two functions. f(x) and g(x) are two functions that can be represented by the arithmetic operations shown below (x). It is important to note that these formulas are only valid when the domains of both functions are the same (or restricted to the same domain).

Addition of functions:

The intersection of the domains of the independent functions is the domain of the sum of two functions. The sum of two functions at a given input is the same as the sum of the independent functions at the same input when both functions are given the same input. i.e.,

        (f + g) (x) =  f(x) + g(x)

Example: 

When f(x) = x2 + 2 and g(x) = x + 1, the equations are as follows:

(f + g)(x) = f(x) + g (x)

= x2 + 2 + x + 1

= x2 + x + 3

Because the domain of each of f(x) and g(x) is the set of all real numbers, R, the domain of (f + g)(x) (which is R ∩ R = R) is R.

Subtraction of function:

This is the intersection of the domains of two independent functions, which is the domain of the difference of two independent functions. It is equal to the difference of two functions at a given input if both functions are independent of each other at the same input. i.e.,

              (f-g) (x) = f(x) – g (x)

Examples:

When f(x) = x2 + 2 and g(x) = x + 1, the equations are as follows:

In other words, (f-g)(x) equals f(x) – g (x)

= x2 + 2 – (x + 1)

= x2 – x + 1

As a result, since each of the functions f(x) and g(x) has as domain the set of all real numbers, R, the domain of (f – g)(x) also has as domain the set of all real numbers, R.

Multiplication of functions:

When two functions are combined, the intersection of their respective domains is referred to as the domain of the product of the two functions. It is equal to the product of two functions at a given input if and only if both functions are independent of each other at that input. The product of functions results in a binomial function, a cubic function, or even a polynomial function, depending on the type of functions involved.

            (f . g) (x) =  f(x) . g (x)

Examples:

When f(x) = x2 + 2 and g(x) = x + 1, the equations are as follows:

(f.g)(x) =  f(x) . g (x)

= (x2 + 2) · (x + 1)

= x3 + x2 + 2x + 2

This is due to the fact that the domain of each of the functions f(x) and g(x) is the set of all real numbers, R, and the domain of (f.g)(x) is R as well.

Division of functions:

When two functions are multiplied together, the quotient’s domain is the intersection of the domains of both of the independent functions. Yet there is one more condition that must be met: we must set the denominator function to be “not equal to zero,” because otherwise the fraction will be undefined if the denominator equals 0. Whenever two functions are used together, their quotient at a given input is equal to their quotient when they are used independently at the same input. i.e.,

                (f / g) (x) = f(x) / g(x),Given that g(x) is not equal to zero. 

Examples:

When f(x) = x2 + 2 and g(x) = x + 1, the equations are as follows:

= (f/g) (x) = f(x)/g(x) 

(x2 + 2) / (x + 1) 

Because the domain of each of f(x) is the set of all real numbers, R; and the domain of g(x) is the set of all real numbers except -1 (as x + 1 is in the denominator, x + 1 ≠ 0 ⇒ x ≠ -1). So the domain of (f / g)(x) is R – {-1}.

Conclusion:

In function algebra, the arithmetic operations of the functions are dealt with in more detail. The domains of two functions must be the same in order to perform any arithmetic operation between them. The operations of the functions, specifically the arithmetic operations, are referred to as the algebra of functions. 

The intersection of the domains of the independent functions is the domain of the sum of two functions. The sum of two functions at a given input is the same as the sum of the independent functions at the same input when both functions are given the same input. i.e.,

        (f + g) (x) =  f(x) + g(x)

This is the intersection of the domains of two independent functions, which is the domain of the difference of two independent functions.

When two functions are combined, the intersection of their respective domains is referred to as the domain of the product of the two functions.

               (f-g) (x) = f(x) – g (x)

The product of functions results in a binomial function, a cubic function, or even a polynomial function, depending on the type of functions involved.

            (f . g) (x) =  f(x) . g (x)

When two functions are multiplied together, the quotient’s domain is the intersection of the domains of both of the independent functions.

Whenever two functions are used together, their quotient at a given input is equal to their quotient when they are used independently at the same input. i.e.,

                (f / g) (x) = f(x) / g(x).