Properties of Inverse Functions

Tutorial on the 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

    graph of function f and its inverse


  6. 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.