Problem 51 - Contemporary Precalculus

Question

Prove that every function \(f\) that has an inverse function \(g\) is one-to-one. I Hint: The proof of the Round-Trip Theorem on page 223 shows that \(f\) and \(g\) have the round-trip properties; use Exercise \(50(\mathrm{a}) .]\)

Step-by-Step Solution

Answer

Question: Prove that every function with an inverse function is one-to-one. Answer: To prove that a function with an inverse function is one-to-one, we showed that for any \(x_1\) and \(x_2\) in the domain of the function, if \(f(x_1) = f(x_2)\) then \(x_1 = x_2\). We used the round-trip properties of the function and its inverse to show this result. It means that the function assigns a unique output to each input, and thus, the function is one-to-one.

Step 1: Define the given functions and their properties

Let \(f\) be a function with an inverse function \(g\). This means that: 1. \(g(f(x)) = x\) for all \(x\) in the domain of \(f\) 2. \(f(g(y)) = y\) for all \(y\) in the domain of \(g\)

Step 2: Assume \(f(x_1) = f(x_2)\) for some \(x_1\) and \(x_2\) in the domain of \(f\)

Let's assume that there are two values, \(x_1\) and \(x_2\), in the domain of \(f\) such that \(f(x_1) = f(x_2)\). We want to show that \(x_1 = x_2\).

Step 3: Apply the inverse function \(g\) to both sides of the equation \(f(x_1) = f(x_2)\)

Apply the inverse function \(g\) to both sides of the equation: \(g(f(x_1)) = g(f(x_2))\)

Step 4: Use the round-trip property of \(f\) and \(g\) to simplify the equation

Now, we can use the round-trip property of \(f\) and \(g\), which states that \(g(f(x)) = x\) for all \(x\) in the domain of \(f\). Thus: \(g(f(x_1)) = x_1\) \(g(f(x_2)) = x_2\)

Step 5: Conclude that \(x_1 = x_2\)

Since \(g(f(x_1)) = x_1\) and \(g(f(x_2)) = x_2\), and we previously found that \(g(f(x_1)) = g(f(x_2))\), we can conclude that: \(x_1 = x_2\)

Step 6: State that the function \(f\) is one-to-one

Since we have shown that for any \(x_1\) and \(x_2\) in the domain of \(f\), if \(f(x_1) = f(x_2)\) then \(x_1 = x_2\), we can conclude that the function \(f\) is one-to-one.

Key Concepts

One-to-One Function

Understanding the concept of a one-to-one function is essential when discussing inverses. A function is considered one-to-one (injective) if different inputs always lead to different outputs. In simpler terms, if you plug two different values into the function, you will get two different results. This property is central to the idea of functions having inverses because only one-to-one functions can have inverses that are also functions.

• If function \( f \) satisfies \( f(x_1) = f(x_2) \), then it must be that \( x_1 = x_2 \).
• This means no two distinct points in the domain map to the same point in the codomain.

For example, the function \( f(x) = 2x + 3 \) is one-to-one because if \( 2x_1 + 3 = 2x_2 + 3 \), it simplifies to \( x_1 = x_2 \). Hence, proving whether a function is one-to-one is often the first step in establishing if it has an inverse.

Round-Trip Theorem

The Round-Trip Theorem is a stepping stone in understanding the nature of inverse functions. It states that if a function \( f \) has an inverse \( g \), then composing them in specific directions will always return you to your starting point.

• In symbols, \( g(f(x)) = x \) for all \( x \) in \( f \)'s domain and \( f(g(y)) = y \) for all \( y \) in \( g \)'s domain.
• This effectively means moving around the function world in a loop: from a point \( x \) through \( f \) to \( g \), and back to \( x \) again.

Consider it as a verification for inverses. If you can return to your origin from \( x \) or \( y \), it proves that \( f \) and \( g \) are indeed inverses. This theorem is instrumental in proving functional properties, like a function being one-to-one.

Function Properties

Functions have various properties that help us understand their behavior and characteristics. Some critical properties include:

• Domain: The set of all possible inputs for the function.
• Codomain: The set of all possible outputs.
• Injective (One-to-One): Different inputs produce different outputs.
• Surjective (Onto): Every element in the codomain is mapped by at least one element in the domain.
• Bijective (One-to-One Correspondence): Both injective and surjective, meaning every element in one set is paired with exactly one element in the other.

Functions need to be bijective to have an inverse. This is because a bijective function ensures that each element leads to one unique counterpart, making the inversion process possible without ambiguity.

Proof Techniques

When proving mathematical properties, particularly in functional analysis, several methods and techniques can be employed. A direct proof is the most straightforward, where you start with known facts or assumptions and logically deduce the conclusion.

• The solution provided follows a direct proof strategy. It begins with the given that \( f \) has an inverse \( g \), and uses their round-trip properties.
• Assume \( f(x_1) = f(x_2) \) and apply the inverse \( g \) to show \( x_1 = x_2 \).

Another common technique is proof by contradiction, where you assume the opposite of what you're trying to prove, and show this assumption leads to an inconsistency. Proof techniques are tools that help us establish the truth in logical and structured ways, empowering us to draw accurate mathematical conclusions.