**Inverse Functions –**

In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a returns the input value given to b. Additionally, this must hold true for every element in the domain co-domain(range) of b. In other words, assuming x and y are constants, if b(x) = y and a(y) = x then the function a is said to be an inverse of the function b.

**Example of Inverse Function –**

Consider the functions a(x) = 5x + 2 and b(y) = (y-2)/5. Here function b is an inverse function of a. We can see this by inserting values into the functions. For example when x is 1 the output of a is a(1) = 5(1) + 2 = 7. Using this output as y in function b gives b(7) = (7-2)/5 = 1 which was the input value to function a.

**Properties of Inverse Functions –**

Two functions f and g are said to be inverses of each other if and only if:

- f and g are both one to one functions. One to One functions map each value in their domain to exactly one value in the co-domain(range). An example of a One to One function is f(x) = x
- The co-domain(range) of f is the domain of g and vice versa

**Note:** Some functions are only invertible for a set of specific values in their domain. In this case both the range and domain of the inverse function are restricted to only those values.

**Composite Functions –**

A composite function is a function whose input is another function. So, if we have two functions A(x), which maps elements from set B to set C, and D(x), which maps from set C to set E, then the composite of these two functions, written as *DoA*, is a function that maps elements from B to E i.e. DoA = D(A(x)).

For example consider the functions A(x) = 5x + 2 and B(x) = x + 1. The composite function *AoB* = A(B(x)) = 5(x+1) + 2.

**Properties of Composite Functions –**

Composite functions posses the following properties:

- Given the composite function
*fog*= f(g(x)) the co-domain of g must be a subset, i.e. proper or improper subset, of the domain of f - Composite functions are associative. Given the composite function
*a o b o c*the order of operation is irrelevant i.e. (a o b) o c = a o (b o c). - Composite functions are not commutative. So
*AoB*is not the same as*BoA*. Using the example A(x) = 5x + 2 and B(x) = x + 1*AoB*= A(B(x)) = 5(x+1) + 2 while*BoA*= B(A(x)) = (5x + 2) + 1.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.