Properties of Inverse Functions
Tutorial on the properties of inverse functions .
The properties of inverse functions are listed and discussed below.
Property 1Only one to one functions have inversesIf g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other. Property 2If f and g are inverses of each other then both are one to one functions.Property 3f and g are inverses of each other if and only ifand (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. Property 4If f and g are inverses of each other thenand the range of f is equal to the domain of g. Example Let f(x) = √ (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 = √ (x  3) and interchange x and y to obtain the inverse f^{ 1} (x) = x ^{2} + 3 According to property 4, The domain of f^{ 1} is given by the interval [0 , + infinity) The range of f^{ 1} is given by the interval [3, + infinity) Property 5If 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) = √ (x  3) and its inverse f^{ 1}(x) = x ^{2} + 3 , x >= 0 Property 6If point (a,b) is on the graph of f then point (b,a) is on the graph of f^{1}.More References and Links to Inverse Functions
