A function is called one-to-one if no two values of x produce the same y. It is a fairly simple definition of one-to-one although it takes an instance of a function which isn't one-to-one to illustrate just what it means.  However Before doing that we have to note that this definition of one-to-one is not actually the mathematically correct definition of one-to-one.  This is similar to the mathematically correct definition it just doesn't employ the entire notation from the formal definition.

Now, let's look at an example of a function which isn't one-to-one.  The function f ( x )= x²  is not one-to-one since both f ( -2) = 4 and f ( 2) = 4 .  In other terms there are two distinct values of x which produce the similar value of y. Note that we can turn f ( x ) = x²  into a one-to-one function if we limit ourselves to 0 ≤ x <∞ .  Sometimes it can be done with functions.

Illustrating that a function is one-to-one is frequently a tedious and often difficult.  For the most of the part we are going to suppose that the functions which we're going to be dealing along with in this section are one- to-one. We did have to talk regarding one-to-one functions though since only one-to-one functions can be inverse functions.