Problem 52 - Discrete Mathematics with Applications

Question

The inverse of \(f\) is unique.

Step-by-Step Solution

Answer

We assumed the existence of two distinct inverses for function f, called g and h, and that both satisfy the inverse properties. By using the definition of inverses, we managed to obtain \(h(x) = g(x)\) for all x in their domain, contradicting our initial assumption. Therefore, the inverse of a function f is unique.

Step 1: Assume two distinct inverses

Let's assume that there are two distinct inverses for function f, which we'll call g and h. We assume that g and h are both inverses of f, which means that for all x in their domain, \[f(g(x)) = x\] and \[f(h(x)) = x\]

Step 2: Use the definition of inverses

We need to use the definition of inverses to make a connection between g and h. Since both g(x) and h(x) are inverses of f, we can substitute one of them into the equation for the other inverse: \[g(f(h(x))) = g(x)\]

Step 3: Apply the inverses properties

Now we will use the properties of inverses on the left side of the equation: \[g(f(h(x))) = g(x)\] Because g is an inverse of f, we know that \(g(f(x)) = x\) for all x. So, applying that to our equation, we have: \[h(x) = g(x)\]

Step 4: Conclude the uniqueness of the inverse

Since the equation \(h(x) = g(x)\) holds for all x in their domain, we can conclude that g and h are not distinct inverses but instead are the same function. This contradicts our initial assumption that there are two distinct inverses for function f. Thus, we have proven that the inverse of a function f is unique.

Key Concepts

Function Composition and its Role in Inverses

When working with inverse functions, function composition is key. Function composition involves substituting one function into another. Specifically, if we have functions \( f \) and \( g \), composing \( g \) with \( f \) means creating a new function expressed as \( f(g(x)) \).
This is like a function inside a function, literally feeding the output of one directly into the other. For inverses, function composition helps verify whether two functions are really inverses of one another.

For example, if \( g \) is indeed the inverse of \( f \), plugging \( g(x) \) inside \( f \) should give you back \( x \), i.e., \( f(g(x)) = x \).
Similarly, plugging \( f(x) \) inside \( g \) should also return \( x \), so \( g(f(x)) = x \).
This relationship needs to hold true for all \( x \) in the domain for \( g \) to truly be \( f \)'s inverse.
In nutshell, function composition confirms the bi-directional nature of inverse functions—each undoes the effect of the other.

Unraveling the Uniqueness of Inverse

The concept of uniqueness arises when dealing with inverses of a function. In simpler terms, a function cannot have two different inverses.
Let's explore this through thought experiments and logic.

Imagine having a function \( f \) and two potential inverses, \( g \) and \( h \).

• Both \( g \) and \( h \) satisfy the properties \( f(g(x)) = x \) and \( f(h(x)) = x \).
• Due to inverse properties, substituting these functions in each other gives \( g(f(h(x))) = g(x) \).
• Because \( g \) is an inverse of \( f \), this simplifies to \( h(x) = g(x) \).

This simplifies, pointing out that \( g \) and \( h \) must be the same function if they're inverses of \( f \).
This logical progression demonstrates that the initial assumption of two distinct inverses leads to a contradiction. Therefore, a function can only have one inverse, reflecting mathematical precision where every operation has one unique undoing.

The Importance of Mathematical Proof

Mathematical proof is a structured argument that follows logical steps to arrive at a conclusion. Proofs are essential in mathematics to verify statements, ensuring they are universally true.
The solution to our problem is a classic example of a proof, showing that an argument holds for all applicable cases.

To prove the uniqueness of an inverse, one needs clear assumptions and logical deductions which are cross-verified. The steps involve:

• Assuming the existence of two inverses and examining their properties.
• Applying function composition and the inverse relation through algebraic manipulation.
• Deriving logical equivalences like \( h(x) = g(x) \) to point out contradictions.

Each step builds on previous ones, reaching a conclusion that can't be refuted.
This methodical reasoning doesn't just confirm the statement but reinforces why it works, giving students a concrete understanding of the delicate balance in mathematics.