Inverse functions and composition of functions

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.

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