One-to-one function: A function is called one-to-one if not any two values of x produce the same y.  Mathematically specking, this is the same as saying,

 f ( x₁ ) ≠ f ( x₂ )

whenever  x₁ ≠ x₂

Thus, a function is one-to-one if whenever we plug distinct values into the function we get different function values. Sometimes it is simpler to understand this definition if we illustrates a function that isn't one-to-one.

 Let's take a look at 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 different values of x that generate the same value of y.  Note down that we can turn f ( x ) = x²  into a one-to-one function if we limit ourselves to 0 ≤ x <∞ .  It can sometimes be done with functions.

Illustrating that a function is one-to-one is frequently tedious and/or difficult.  For the most part we are going to suppose that the functions which we're going to be dealing with in this course are either one-to-one or we have limited the domain of the function to get it to be a one-to-one function.

Now, let's formally define just what inverse functions are.