Here are 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 specified it by

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

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

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

The notation which we use really based upon the problem.  In most of the cases either is acceptable.

For the two functions which we started off this section along with we could write either of the following 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

Now, be careful along with the notation for inverses. The "-1" is not an exponent in spite of the fact that is certain does look like one! While dealing with inverse functions we've got to keep in mind that

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