Inverse Functions : In the last instance from the previous section we looked at the two functions

  f ( x ) = 3x - 2 and g ( x ) = x /3+ 2/3 and saw that

( f o g ) ( x ) = ( g o f ) ( x ) = x

and as noted in that section it means that there is a nice relationship among these two functions.  Let's see what that relationship is.  Assume the following evaluations.

f ( -1) = 3( -1) - 2 = -5

⇒         g ( -5) = -5 /3+ 2/3 = -3/3 = -1

g ( 2) = 2/3 + 2/3 =4/3

⇒         f ( 4 /3) = 3( 4/3 ) - 2 = 4 - 2 = 2

In the first case we plugged x = -1 in f (x) and got a value of -5.  Then we turned around and plugged x = -5 into g (x) and got a value of -1, the number which we started off with.

In the second case we did something same.  Here we plugged x = 2 into g ( x ) and got a value of 4/3, we turned around & plugged this into f ( x ) and got a value of 2, that is again the number that we begun with.

Note that we actually are doing some function composition here. The first case is,

 ( g o f ) ( -1) = g [f ( -1)]= g (-5) =-1

and the second case is,

 ( f o g ) ( 2) = f [g ( 2)] =f(4/3)=2

Note that these both agree with the formula for the compositions which we found in the previous section.  We get back of function evaluation the number which we originally plugged into the composition.

Thus, just what is going on here?  In some of the way we can think of these two functions as undoing what the other did with number.  In the primary case we plugged x = -1 into f ( x ) and then plugged the result from this function evaluation back into g (x ) and in some way g (x ) undid what f ( x ) had done to x = -1 and gave us back the original x which we started with.

Function pairs which exhibit this behavior are called inverse functions. Previous to formally defining inverse functions & the notation which we're going to use for them we have to get a definition out of the way.