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Inverse and Composition of Function

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In mathematics the Inverse Function is considered as a function that undoes another function. If ‘g’ is a called a function then the inverse of the function ‘g’ is denoted as g-1.
The theorem for composition of inverse Functions is stated as follows-
Suppose that there are two Functions named ‘f’ and ‘g’, this would be invertible functions such that the compositions of these functions f o g will be well defined. Mathematically this can be written as shown below-
(f o g)-1 = g-1 o f-1,
To have a better understanding about the composite and inverse functions the following would be helpful.
The function composition always works from right to the left in mathematics. Here (f o g)-1 is the reverse of the process (f o g), this can be understood better by considering the functions in a different way.
Consider the function ‘g’ as putting that function on one's socks, then putting that on one's shoes. As the reverse process of (f o g) is the (f o g)-1 which is taking off one's shoes (that is f-1) and followed by taking off one's socks which is the function g-1.
To find inverse of a function there are a number of ways in mathematics and mostly the inverse of a function is preferred with the help of graphs and the algebraic operations. In other words inverse functions re defined as:
“For each Ordered Pair (x, y) in a function the denotation be ‘g’, there would be an another ordered pair (y, x) definitely in the inverse function.”
The composition of a function can be understood by the following statement:
“Suppose a function ‘g’ that operates on the function f (x) then the composition of the function would be written as [g f (x)]. This function can be renamed as h (x) = [g f (x)]”.

Composition of Function

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Composition of function is the result of one function in the form of other function. This is expressed by the small Circle. Suppose that there are two Functions ‘f’ and ‘g’, Then the composition of function ‘f’ to ‘g’ is denoted as (g o f) (x) or sometimes as g (f (x). This can not be denoted by a simple dot other wise the meaning would be the produce in spite of composition.
If g (x) is given in the problem then composition of Functions is denoted as f (g(x)) = f o g (x). Here it should be noticed that f (g(x)) is not equal to the g (f (x)).
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 composition of 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.