Properties of Inverse Functions

The properties of inverse functions are listed and discussed below.

1. Only one to one functions have inverses
   
   If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other.
   
   
2. If f and g are inverses of each other then both are one to one functions.
   
   
3. f and g are inverses of each other if and only if
   
   (f _o g)(x) = x , x in the domain of g
   
   and
   
   (g _o f)(x) = x , x in the domain of f
   
   Example
   
   Let f(x) = 3 x and g(x) = x / 3
   
   (f _o g)(x) = f( g(x) ) = 3 ( x / 3 ) = x
   
   and g _o f)(x) = g( f(x) ) = (3 x) / 3 = x
   
   Therefore f and g given above are inverses of each other.
   
   
4. If f and g are inverses of each other then
   
   the domain of f is equal to the range of g
   
   and
   
   the range of f is equal to the domain of g.
   
   Example
   
   Let f(x) = sqrt (x - 3)
   
   The domain of f is given by the interval [3 , + infinity)
   
   The range of f is given by the interval [0, + infinity)
   
   Let us find the inverse function
   
   Square both sides of y = sqrt (x - 3) and interchange x and y to obtain the inverse
   
   f ^-1 (x) = x ² + 3
   
   According to property 5,
   
   The domain of f ^-1 is given by the interval [0 , + infinity)
   
   The range of f ^-1 is given by the interval [3, + infinity)
   
   
   
5. If f and g are inverses of each other then their graphs are reflections of each other on the line y = x.
   
   Example
   
   Below are the graphs of f(x) = sqrt (x - 3)
   and
   its inverse f ^-1(x) = x ² + 3 , x >= 0
   
   
   
   
6. If point (a,b) is on the graph of f then point (b,a) is on the graph of f^-1.