Composite Functions

In this article, we you will learn about composite functions, it means that a function can become input of another function. Before you learn about the composite functions, you must be familiar with the arithmetic of functions which means given two function , you will be able to perform basic arithmetic operations on the function itself. Identifying Domain

Sometimes the function expression is given, but domain is not specified. In such cases, you must identify the domain of given function. Suppose

   

is a function whose domain is which is “all real numbers”, because the function is true for all real numbers. Another function

   

has a domain which must have all numbers that are otherwise the function is invalid. Therefore, contains all numbers in the interval .

Why this is important ? because when we perform arithmetic operations on two or more functions the domain of new function is set of all real numbers that belong to both and , that is, .

Arithmetic Operations on Functions

Normal arithmetic operations are possible on functions too. If and are two functions, then there are four operations possible on these functions.

1. Sum
2. Difference
3. Product
4. Quotient

Now we discuss about each of these operations.

Sum of functions

The sum of functions is

   

The expressions of both functions are added together to form a new function. For example, if and , then

   

Difference of functions

The difference of functions is

   

The expressions of function is subtracted from expression of function to form a new expression for . For example, if and , then

   

Product of functions

The product of the functions is

   

The expressions of function and is multiplied to get the new product expression of . Each term is of is multiplied with each term of . For example, if and are two expressions, then

   

Quotient of functions

The quotient of functions is obtained by dividing two functions, which is

   

The functions are divided to get the quotient, however, there is one condition, that is, , otherwise the quotient of function will be “divide by 0” which is “undefined“.

Therefore, if and , then

   

The domain of , , is set of all reals numbers that are common to domain of and which is except where for .

Composite Functions

The composite functions or the idea of composition of functions is simple. Suppose there are two functions, and , then if function becomes input for function , it is called “composition of function with function or simply a composite function.

Let us try to understand this with an example, suppose Nancy work as a maid, and get paid amount every week and after paying taxes she receives only of her earnings. Each week she spends for grocery from her earning . The total earning of Nancy after paying taxes can be defined by function . Her savings after expense of can be defined as function .

   

If her gross earning is , then her saving would be

   

Therefore, total saving after paying taxes and expenses of is .

What is the domain of a composite function?

If is a composite function such that . Then the domain of composite function must be

1. meaning “ must be in the domain of .
2. meaning “ must be in the domain of .

Therefore, while finding the domain of a composite function, we must first exclude all values of than can make the function “invalid”. Also, if happens to be valid, that is, , then must be valid value for the function . If happens to invalid value, then both and must be excluded from the domain of composite function .

Decomposing Functions

Like composing two functions, it is possible to decompose a function because we know that “composition of two functions” creates a new function. Consider the following example,

   

Here we can clearly see that there are two functions involved in the expression. Therefore, where which implies that .

We can write the function and as composition of function where and .