1 - Click on the button "click here to
start" and MAXIMIZE the window obtained.

2 - On the left panel click on "function
f".

3 - Examine the set of points (in blue)
representing function f. Function f is represented by the set of ordered pairs
as follows:

f
= {(-2.5 , -4.0) , (-2.0 , -3.0) , (-1.5 , -2.0) , (-1.0 , -1.0) , (0.0 , 0.5) ,
(1.0 , 1.0) , (2.0 , 1.5) , (3.0 , 2.0) , (4.0 , 2.5)}

4 - Click ( with
the mouse) on any of the points of the graph of f. A point (in red representing)
an ordered pair of the inverse function appears. Examine the coordinates of the
point (blue) in the graph of f and the coordinates of the point (in red) in the
graph of its inverse. What do you notice?

5 - Click on all points of f so that all
corresponding point in the graph of the inverse appear and take note of all the
ordered pairs representing the inverse.

6 - Take any point on the graph of f and its
corresponding point on the graph of the inverse. Compare their position with
respect to the line y = x (in green). What do you notice? Show that the midpoint
of the two points is on the line y = x and show that the line through the two
points is perpendicular to the line y = x. Conclusion?

7 - Find the domain and range of function f (blue).

8 - Find the domain and range of the inverse
of f (red). Describe the relationship of the domain and range of f to the domain and range of its inverse.

__II - Which functions do not have an inverse?__

9 - We now use another function g. On the
left panel, click on "function g". Click on all the blue points making
the graph of function g. Is the graph obtained (in red) that of a
function? (Hint: Use the vertical line test or examine the ordered pairs
defining function g to answer this question).

10 - Examine function g to find out why it
does not have an inverse. More on one-to-one functions.

__Matched Exercises__

**Exercise 1:**

a) Find the domain and range
of function f defined by

f = {(-4,2),(-3,1),(0,5),(2,6)}

b) Find the inverse function of f and its domain and range.

**Exercise 2:** Which of these functions do not have an
inverse?

f = {(-1,2),(-3,1),(0,2),(5,6)}

g = {(-3,0),(-1,1),(0,5),(2,6)}

h = {(2,2),(3,1),(6,5),(7,1)}

__Answers to Above Matched Exercises__

**Exercise 1:**

a) domain of f = {-4,-3,0,2} and range of f = {2,1,5,6}

b) inverse of f = {(2,-4),(1,-3),(5,0),(6,2)}

domain of inverse of f = {2,1,5,6} and range of inverse of f = {-4,-3,0,2}.

**Exercise 2:**
Functions f and h do not have inverses.

More links and references related to the inverse functions.