Properties of Inverse Functions

Inverse functions are functions that "reverse" the effect of the original function. Given a function ( f: A \rightarrow B ), an inverse function ( f^{-1}: B \rightarrow A ) exists if ( f ) is a bijection, which means it is both injective (one-to-one) and surjective (onto).

Basic Properties of Inverse Functions

1. Existence: An inverse function exists only for bijective functions.
2. Uniqueness: If an inverse function exists, it is unique.
3. Composition: The composition of a function and its inverse yields the identity function: ( f(f^{-1}(x)) = x ) and ( f^{-1}(f(x)) = x ).
4. Domain and Range: The domain of ( f ) is the range of ( f^{-1} ) and vice versa.
5. Symmetry: The graph of ( f^{-1} ) is the reflection of the graph of ( f ) across the line ( y = x ).

Table of Differences and Important Points

Property	Function ( f )	Inverse Function ( f^{-1} )
Domain	Set ( A )	Set ( B )
Range	Set ( B )	Set ( A )
Notation	( f(x) )	( f^{-1}(x) )
Graph	( y = f(x) )	Reflected across ( y = x )

Formulas Involving Inverse Functions

1. Composition: ( (f \circ f^{-1})(x) = f(f^{-1}(x)) = x ) and ( (f^{-1} \circ f)(x) = f^{-1}(f(x)) = x ).
2. Derivative: If ( f ) is differentiable and ( f^{-1} ) exists, then ( (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))} ), provided ( f'(f^{-1}(y)) \neq 0 ).

Examples to Explain Important Points

Example 1: Existence and Uniqueness

Consider the function ( f(x) = x^3 ). This function is bijective since each value of ( x ) produces a unique value of ( f(x) ), and every possible value of ( f(x) ) is produced by some value of ( x ). Its inverse is ( f^{-1}(x) = \sqrt[3]{x} ), which is unique.

Example 2: Composition

Let ( f(x) = 2x + 3 ) and ( f^{-1}(x) = \frac{x - 3}{2} ). Then:

( f(f^{-1}(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x )

( f^{-1}(f(x)) = f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = x )

Example 3: Symmetry

The function ( f(x) = \ln(x) ) has an inverse ( f^{-1}(x) = e^x ). The graph of ( f^{-1}(x) ) is the reflection of the graph of ( f(x) ) across the line ( y = x ).

Example 4: Derivative of an Inverse Function

If ( f(x) = \ln(x) ), then ( f'(x) = \frac{1}{x} ) and ( f^{-1}(x) = e^x ). Using the formula for the derivative of an inverse function:

( (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))} = \frac{1}{\frac{1}{e^y}} = e^y )

This result is consistent with the known derivative of the exponential function.

Understanding the properties of inverse functions is crucial for solving problems in calculus, particularly when dealing with integrals and derivatives of inverse trigonometric functions and logarithmic functions. It is also important in algebra when finding the inverse of a given function to determine if two functions are inverses of each other.