# 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.