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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.

- Sum
- Difference
- Product
- 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

- meaning “ must be in the domain of .
- 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 .