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Tuesday, November 27, 2007

Definition of a one-to-one function

Explore the concept of one-to-one function using an applet. This concept is necessary to understand the concept of inverse function. Several functions are explored graphically using the horizontal line test. The exploration is carried out by changing parameters a, b and c included in these functions. Examples of analytical explanations are, in some cases, provided to support the graphical approach followed here. Some definitions are reviewed so that the explorations can be carried out without difficulties.

Definition of a Function: A function is a rule that produces a correspondence between the elements of two sets: D ( domain ) and R ( range ), such that to each element in D there corresponds one and only one element in R.

Definition of a one-to-one function: A function is a one-to-one if no two different elements in D have the same element in R.
The definition of a one to one function can be written algebraically as follows:

Let x1 and x2 any elements of D

A function f(x) is one-to-one

I - if x1 is not equal to x2 then f(x1) is not equal to f(x2)

OR the contrapositive of the above

II - if f(x1) = f(x2) then x1 = x2. This last property can be useful as we shall see later in the tutorial.
 
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