A man can be as great as he wants to be. If you believe in yourself and have the courage, the determination, the dedication, the competitive drive and if you are willing to sacrifice the little things in life and pay the price for the things that are worthwhile, it can be done. – **Vince Lombardi**

# Lecture - 6 Relations and Functions

**INVERTIBLE FUNCTIONS**

If f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible. This fact significantly helps for proving a function f to be invertible by showing that f is one-one and onto, especially when the actual inverse of f is not to be determined.

A function f : X → Y is defined to be invertible, if there exists a function g : Y → X such that gof= I_X and fog=I_Y. The function g is called the inverse of f and is denoted by f^{-1}.

OR

If a function f defined by f : X → Y and the inverse function defined by f^{-1}:Y \rightarrow X then f^{-1}of(x)=x and fof^{-1}(x)=x.