Part I: Definition, properties of inverse functions
1  Definition
Let function f be defined by the set of ordered pairs as follows:
f = {(1,0),(4,5),(6,9)}
By interchanging the first and second coordinates of each ordered pair in f, we can define the inverse function of f as follows:
f^{ 1} = {(0,1),(5,4),(9,6)}
Notes 1: The domain of f is equal to the range of f^{ 1} and the range of f is equal to the domain of f^{ 1}.
Notes 2:
 f(f^{ 1}(0)) = f(1) = 0
 f(f^{ 1}(5)) = f(4) = 5
 f(f^{ 1}(9)) = f(6) = 9
and
 f^{ 1}(f(1)) = f^{ 1}(0) = 1
 f^{ 1}(f(4)) = f^{ 1}(5) = 4
 f^{ 1}(f(6)) = f^{ 1}(9) = 6
and in general
f(f^{ 1}(x)) = x , x in the domain of f^{ 1}
f^{ 1}(f(x)) = x , x in the domain of f
The inverse function, denoted f^{ 1}, of a onetoonefunction f is defined as
f^{ 1}(x) = {(y,x)  such that y = f(x)}
f does not have an inverse if it is not a onetoone function.
You might want to go through an interactive tutorial using a java applet on definition of the inverse function.
Part II: Interactive Tutorial, Graphical investigation of the properties of the inverse functions
From the definition of the inverse function given above, if point (a,b) is on the graph of function f then point (b,a) is a point on the graph of f^{ 1}. But points (a,b) and (b,a) are reflection of each other on the line y = x. This property and its consequences on the graph of a function and its inverse are investigated in part II.
This applet allows you to plot the graph of any of the
following functions (in blue):
 f(x) = a*(b(x+c))^{2} + d, for x >= c
 f(x) = a*(b(x+c))^{3} + d
 f(x) = a*exp(b(x+c)) + d
 f(x) = a*sin(b(x+c)) + d, for Pi/2 <= b(x+c) <= Pi/2
and its inverse (in red). You can also plot your own data points for comparison.
You can change any of the 4 parameters a, b, c ,d to shift, compress or stretch the graph of the selected function and see its effect on the graph of the inverse function.
Click on the button above "click here to start" and MAXIMIZE the window obtained.
 Only one to one functions have inverse functions. Check graphically (horizontal line test) and analytically that all 4 functions defined above are one to one functions.
 Select a function and set a=1, b=1, c=0 and and change d (vertical translation of f(x)).
what are the effects on the inverse function? explain these effects from what you
know on inverse functions.
 set a=1, b=1, d=0 and and change c (horizontal translation of f(x)).
what are the effects on the inverse function? explain these effects from what you
know on inverse functions.
 set a=1, d=0, c=0 and and change b (horizontal scaling of f(x)).
what are the effects on the inverse function? explain these effects from
what you know on inverse functions.
 set b=1, d=0, c=0 and and change a (vertical scaling of f(x)).
what are the effects on the inverse function? explain these effects from
what you know on inverse functions.
 take any of the 4 functions in the list (left panel, top), find
its inverse,make a table of points both for f(x) and its inverse function
and plot them to compare.
More links and references related to the inverse functions.
Find inverse of exponential functions
Applications and Use of the Inverse Functions
Find the inverse of a Function  Questions
Find the Inverse Function (1)  Tutorial.
Find the Inverse Function (2)  Tutorial
Definition of the Inverse Function  Interactive Tutorial
Find Inverse Of Cube Root Functions.
Find Inverse Of Square Root Functions.
Find Inverse Of Logarithmic Functions.
Find Inverse Of Exponential Functions.
Properties of Inverse Functions
