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.
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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
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.
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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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