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Composite Functions Composite Functions To understand about the concept of composite functions, first we must know that what is a function. The study of function is the study of mathematics, which help us to solve many equations with single variable or more than one variable. We say that the function is any mathematical expression which helps to change the value of one variable into another. It gives us the set pattern which helps us to change the value of the number in the same way. Now let us say that the composite function is the combination of any two functions, where we will apply the first function and get the solution for the first function. This value of the first function will help us to get the value of the second function, by simply putting the value of first function into another function. We also call it the function of the function. The value of the first function is extracted from the function. Let us consider two simple functions such that they represent the following expressions :

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F(x) = 2x + 5 and another function is g ( x) = 3 x^2 To represent the composite function relation between the two functions, we write the composite function as follows: f ? g ( x). This small circle between the f and g represent the composite function of f and g. It is used to represent f ( g ( x )), it simply means that first we are going to work on the g( x ) function and then we will get the result and this result is used to work on the function f. It will be clearer with the following example: Let us write the function: f ? g ( 5 ). We know that g ( x ) is equal to 3 x ^2, so we say that first we will find the value of g ( 5), by putting the value of x = 5, in the function and we will get G( 5) = 3 * 5 ^2 = 3 * 5 * 5 = 75 . Now we say that the value of g ( 5) is 75, so we will now place the value of g ( 5) = 75 in the function f, so we will find the value of f( 75) by using the function f( x ) = 2x + 5 Thus here we will put the value of f(x) , by putting the value of x = 75. We get : f ( x) = 2 * 75 + 5 Or we write f ( 75 ) = 150 + 5 = 155. Now let us consider another pair of functions f and g such that : f( x) = 3x + 3 and g( x ) = 2 x ^2 + 2 To find the value of f ( g ( x) ) , we will first find the value of g(x) and then the value so attained will be used to find the value of the function f (x) Such relations formed with the functions are called the composite functions

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A function can be composite with the self and is written as (f o f) (x) that is f (f (x)). Any function works only under its domain. A domain is the set of all the values that are covered by the function. To find the domain of the composite function one needs to remind some Point. Suppose that there is a composite function g (f (x)) the: First make sure that the domain of the function f (x) should be obtained first. Then the domain of the other function ‘g’ is defined according to the first function f (x). While the evaluation of a function if a function f (x) is given to us and asked to find the value of the function f(2), we replace ‘x’ with 2 in the function. Similarly if our notation for the composite functions is f (g(x)) and we have to calculate the ‘f’ function then we will replace the ‘x’ with g (x) everywhere. This is not so different from the evaluation process.

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Composite Functions