Real Analysis/Inverse Functions

In this chapter, we will be formalizing the definition and the intuitive behaviors of an inverse function. In earlier mathematics, you may have been taught a cursory amount on the subject, such as their reflection on the $$ line or a list of functions and their inverses. You may have been taught the inverse trigonometric functions as a hand-wavy method to get an angle from the lengths of an imaginary triangle. You may have understood that the nth root is the inverse of an nth power. Here, we will formalize it—give it meaning and thus give it intrigue. Yet, many theorems related to inverse functions will not be immediately useful or understandable unless other concepts have been constructed beforehand. Thus, sections of this chapter may contain notices for any concepts needed beforehand.

Construction
The construction of the inverse function can be done in two methods. The first method is to simply define the inverse function outright. We will not define it in that manner in this wikibook. Instead, we will define it as a consequence of a new operation. We will define inverse functions in this manner because it assumes less concepts and gives us a new operator to work with (It will be used in a very important concept in this chapter). This section of this chapter will express how we will follow the second method in order to define it.

Proof
The proof naturally follows from the definition, if set up correctly.

We also declared that the corollary, the biconditionality of the previous theorem, is true as well; let's prove that too.

Notation
There is a small list of naming conventions associated with inverse functions that are designed to make things less confusing. Typical ordered pair names, such as $ƒ$ or $(a, b)$ should be used to reflect the inverse function's nature of reversing ordered pairs. For example, $(b, a)$ or $ƒ$ complements the naming convention of $ƒ^{-1}$ and $$ respectively.

Theorems
The curious aspect of inverse functions is its ability to create new definitions of known functions and theorems by, in a sense, reversing known functions and theorems. We will first start with proving very simple consequences from the theorem defining inverse functions, then we will move to a curious proof relating to algebra, then we will advance into calculus by proving differentiation properties.

Basic Properties
If it was not obvious, there one seemingly necessary property that ought be easily derived from the definition.

Proof
The proof relies on an application of function and inverse function definitions.

The results of this proof seems to justify its notation; simply "cancel" out inverses of functions if the functions are composed on each other as if it was multiplication of a variable $(a, ƒ(a)) = (-a, ƒ(a))$ and its inverse $(ƒ(a), a) and (ƒ(a), -a)$ It also proves our intuitive notion that we ought have a way to undo mathematical operations.

Algebra
Now that we have proven the intuitive understanding that the inverse function is the opposite of the function, we can work on the second part necessary to justify algebra, which is that of algebraic reversibility. Algebraic reversibility is the mathematical concept closely related to "if and only if"—the ability to apply an operation that can be reversed by applying the opposite operation. We will prove this concept now.

Proof
This proof will rely only on applying the definition of an inverse function.

Continuity
We will begin by proving that an inverse function is continuous if the original function is continuous. Why are we proving this as opposed to discontinuous functions? Well, on the chapter on limits, we discussed that discontinuity is the not special case. If it is not the special case, it usually has fewer properties to work with and does not need further expansion. THus, we will focus on continuity preservation. As a notice, this theorem requires proving some parts and combining it together. This will mean that the theorem itself will be very long. The theorem itself is written below.

Note that from the inverse function definition we have already proved that all inverse functions $$ are one-one. From there, you might be a little daunted at the process you would have to undergo. However, this is a textbook so we will be your guide.

First, we will need to prove a separate theorem that will be instrumental in piecing the proof together. This theorem is below.

To prove this, we will use the intermediate value theorem.



From this theorem, we will use it to prove our initial theorem regarding continuity. As usual, we will focus on increasing in our proof, as we can emulate decreasing functions by negating the given function. The proof is an epsilon-delta proof.

"The Reciprocal Definition" Property
This following definition that the concept of the inverse function provides is for a derivative definition, in which the derivative of the inverse is known.

As always, this theorem works if you reverse whether the inverse function is inside or outside. We will not be discussing a proof for now. Why? The proof requires both continuity - something we haven't even verified to exist for inverses yet - and differentiation - something we also haven't verified to work for inverse functions. However, we will discuss the intuition behind this property, which we will show below if this property is not intuitive.

Differentiation
Earlier on in the section on inverse functions, we have readily declared that the formula for calculating the derivative of an inverse function, in relationship to the derivative of the original function, as $$. However, it contained a caveat that $ƒ^{-1}$ The proof should be readily inferred from the formula itself (Hint: Substitute $ƒ$). We shown in that section that this can be derived from an application of Chain Rule with an appendix that it actually isn't a proof. We stated that it isn't because we have not worked out whether differentiation works for inverse functions. Well, since we proved that continuity works if the original function is continuous, we now need to check if differentiation also works (since differentiation requires continuity).

Exercises

 * 1) Prove that all periodic functions will not have an inverse function.
 * 2) Prove that all non-restricted even functions $a &ne; b$ will not have an inverse function.