Properties of Inverse Functions

Tutorial on the properties of inverse functions .

The properties of inverse functions are listed and discussed below.

Property 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.

Property 2

If f and g are inverses of each other then both are one to one functions.

Property 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.

Property 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) = √ (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 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) = √ (x - 3)
and
its inverse f^-1(x) = x ² + 3 , x >= 0
graph of function f and its inverse

Property 6

If point (a,b) is on the graph of f then point (b,a) is on the graph of f^-1.