Inverse Function Definition

The inverse function definition is explored using java applets. The conditions under which a function has an inverse are also explored.

I- Explore the Definition of the Inverse

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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.

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