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Aim for the moon. If you miss, you may hit a star. – **W. Clement Stone**

# Lecture - 7 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 â†’ Y is defined to be invertible, if there exists a function g : Y â†’ 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 â†’ Y and the inverse function defined by \( f^{-1}:Y \rightarrow X \) then \(f^{-1}of(x)=x\) and \(fof^{-1}(x)=x\).