Definition of inverse functions :  Given two one-to-one functions f ( x ) and g ( x ) if ( f o g ) ( x ) = x  AND  ( g o f ) ( x ) = x

then we say that f ( x ) & g ( x ) are inverses of each other. More particularly we will say that g ( x ) is the inverse of f ( x ) and denote it by

                                                g ( x ) = f ^-1 ( x )

Similarly we could also say that f ( x ) is the inverse of g ( x) and indicate it by

                                              f ( x ) = g ^-1 ( x )

The notation that we employ really based upon the problem.  In most of the cases either is acceptable. For the two functions which we begun this section along with we could write either of the given two sets of notation.

f ( x ) = 3x - 2                                                     f ^-1 ( x ) = x/3 +2/3

g ( x ) = x/3 + 2/3                                                g ^-1 ( x ) = 3x - 2

 Be careful along with the notation for inverses. The "-1" is NOT an exponent in spite of the fact that is sure does look like one! While dealing along with inverse functions we've got to remember that

                                                f^-1(x)   ≠1/f(x)

It is one of the more common mistakes which students make while first studying inverse functions.