Problem 1 - Algebra 2 and Trigonometry

Question

Taylor said that if \((a, b)\) is a pair of a one-to-one function \(f,\) then \((b, a)\) must be a pair of the inverse function \(f^{-1} .\) Do you agree with Taylor? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Taylor is correct because the inverse function of a one-to-one function reverses its pairs.

Step 1: Understanding One-to-One Functions

A function is one-to-one (injective) if every element of the domain is paired with a unique element of the codomain. This means that no two different inputs can map to the same output.

Step 2: Defining the Inverse Function

An inverse function, denoted as \( f^{-1} \), essentially reverses the pairing of a function \( f \). If a function is one-to-one, then its inverse can be uniquely defined, where each pair \((a, b)\) of the original function maps to a pair \((b, a)\) in the inverse function.

Step 3: Verifying Taylor's Statement

Consider a one-to-one function \( f:\{a, c\} \to \{b, d\} \) with pairs \((a, b)\) and \((c, d)\). By the definition of an inverse function and since \( f \) is one-to-one, the inverse \( f^{-1} \) will have pairs \((b, a)\) and \((d, c)\). This shows that Taylor's statement is true.

Step 4: Conclusion and Justification

Since the function \( f \) is one-to-one, each output is paired with exactly one input. Therefore, the inverse function \( f^{-1} \) will reverse each pair in \( f \), confirming that if \((a, b)\) is in \( f \), then \((b, a)\) will be in \( f^{-1} \). Taylor is correct.

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is where each element of the domain is paired with a unique element in the codomain. This means that no two different inputs lead to the same output. Think of it like a unique identifier where each input has its own distinctive output.

For example, consider a function where the domain \{x, y\} maps to a codomain \{u, v\} such that \(x \, \rightarrow \, u\) and \(y \, \rightarrow \, v\). Since \(x\) and \(y\) map to distinct outputs, this function is one-to-one.

In mathematical terms, for a function \( f\) to be one-to-one, it implies that \[ f(a) = f(b) \Rightarrow a = b. \]

This ensures that each pair of elements is distinct and can uniquely identify the relationship between domain and codomain.

Injective Function

An injective function is a formal way to express a one-to-one function. The word 'injective' emphasizes the 'injection' from each element in the domain into a unique element in the codomain, ensuring no overlap or duplication in the output values.

Injective functions are ideal when precision is key, like assigning a student ID to each student. Every student must have a distinct ID, much like each domain element must map to a distinct codomain element.

In algebra symbols: if \( f(x_1) = f(x_2) \, \Rightarrow x_1 = x_2 \).
This property confirms that an injective function's outputs are unique and individual to each input, securing a precise one-to-one mapping.

Function Pairs

Function pairs refer to the structured relationships between elements in the domain and codomain of a function, often represented as ordered pairs \( (a, b) \). Here, \(a\) belongs to the domain and \(b\) to the codomain.

When analyzing inverse functions, understanding these pairs is crucial. If \( (a, b) \) is in a function, its inverse pair is \( (b, a) \) in the inverse function. It's like flipping your route: if you travel from home \(a\) to school \(b\), the inverse would consider traveling from school back to home.

In the context of inverse functions, function pairs showcase the reversible nature of the relationship, provided, of course, that the function is one-to-one. This ensures every output can trace back to its original input.

Domain and Codomain Relationship

The domain and codomain relationship in a function determines how values are mapped and paired. The domain consists of all possible inputs, while the codomain contains potential outputs. How these sets interact is essential, especially for determining inverses and ensuring the injective property.

In a one-to-one function, every domain element maps uniquely into the codomain. This explicit mapping supports the creation of an inverse function.

For instance, if function \( f:\{1, 2\}\rightarrow\{a, b\} \) has pairs \( (1, a) \) and \( (2, b) \), then each input and output must match distinctly. The precision ensures the inverse function can flawlessly map back, such as \( f^{-1} :\{a,b\} \to \{1,2\} \) with pairs \( (a, 1) \) and \( (b, 2) \).

This relationship is foundational for verifying the correctness of function pairs, particularly in discussing inverse functions and their properties.