Properties of Inverse Functions
Tutorial on the properties of inverse functions .
The properties of inverse functions are listed and discussed below.
 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.
 If f and g are inverses of each other then both are one to one functions.

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.

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)

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

If point (a,b) is on the graph of f then point (b,a) is on the graph of f^{1}.
More links and references related to the inverse functions.

