Why a Function Must Be One-to-One to Have an Inverse Function

Understanding why a function must be one-to-one (injective) to have an inverse function is a crucial concept in mathematics, particularly in Mathematical Analysis, Functional Analysis, and Linear Algebra. A function ( f: A to B ) is injective if each element in the domain ( A ) maps to a unique element in the codomain ( B ). This requirement ensures that the inverse function ( f^{-1}: B to A ) can be defined uniquely and consistently.

The Concept of One-to-One

Definition of One-to-One

A function ( f: A to B ) is one-to-one if for every pair of distinct elements ( x_1, x_2 in A ), the images under ( f ) are also distinct. Mathematically, this means:

( f(x_1) f(x_2) ) implies ( x_1 x_2 )

It is important to note that an injective function must pass the Horizontal Line Test; no horizontal line can intersect the graph of the function at more than one point.

Introduction to Inverse Functions

Definition of Inverse Function

The inverse function ( f^{-1}: B to A ) is a function such that:

( f^{-1}(f(x)) x ) for all ( x in A )

( f(f^{-1}(y)) y ) for all ( y in B )

A key requirement for the existence of an inverse function is that the original function ( f ) must be bijective, meaning it is both injective and surjective.

Why One-to-One is Necessary

An injective function is necessary for the inverse function to be well-defined because it ensures that each element in the codomain ( B ) has at most one element in the domain ( A ) that maps to it. If a function is not one-to-one, there are at least two different inputs ( x_1 ) and ( x_2 ) in the domain such that ( f(x_1) f(x_2) ). This creates ambiguity when trying to define the inverse function.

Ambiguity in Mapping

Suppose ( f(x_1) f(x_2) ) for ( x_1 eq x_2 ). When we attempt to find ( f^{-1}(f(x_1)) ), it could correspond to either ( x_1 ) or ( x_2 ). This ambiguity means the inverse function cannot produce a unique output for a given input.

Not a Function

Since a function must assign exactly one output for each input, the inverse function would fail to satisfy this condition if multiple inputs map to the same output. Therefore, if a function is not one-to-one, the inverse cannot be well-defined and fails to be a function.

Example

Consider the function ( f(x) x^2 ) defined for all real numbers. This function is not one-to-one because:

( f(2) 4 ) and ( f(-2) 4 )

Suppose we attempt to define an inverse function:

( f^{-1}(4) 2 ) and ( f^{-1}(4) -2 )

This ambiguity shows that ( f^{-1} ) cannot be defined uniquely, confirming that ( f(x) x^2 ) does not have an inverse.

Conclusion

In summary, a function needs to be one-to-one to ensure that each output corresponds to exactly one input. This allows for the definition of a unique inverse function. If a function is not one-to-one, the inverse cannot be well-defined, which is why injectivity is a crucial property for functions that have inverses.