Problem 4 - Algebra A Combined Function

Question

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. \(r=\\{(1,2),(3,4),(5,6),(6,7)\\}\)

Step-by-Step Solution

Answer

The function is one-to-one; its inverse is \( r^{-1} = \{(2,1),(4,3),(6,5),(7,6)\} \).

Step 1: Define a One-to-One Function

A function is one-to-one if each input has a unique output, which means no two different inputs map to the same output. To check if a function like the set \( r = \{(1,2),(3,4),(5,6),(6,7)\} \) is one-to-one, examine each pair of inputs and their corresponding outputs.

Step 2: Examine the Given Function Set

Look at the set \( r = \{(1,2),(3,4),(5,6),(6,7)\} \). Check if any two pairs have the same second element. In this case, all pairs have unique outputs (2, 4, 6, and 7), so the function is one-to-one.

Step 3: Verify by Switching Coordinates

Since the function is one-to-one, we can find its inverse by switching the first and second elements in each pair. Thus, the inverse function is \( r^{-1} = \{(2,1),(4,3),(6,5),(7,6)\} \).

Key Concepts

Function Mapping

The concept of function mapping lies at the heart of understanding functions. Function mapping refers to the way each input is paired with an output. Think of it as setting a rule or relationship between a set of inputs and outputs. In the exercise provided, the function is defined by the set \( \{(1,2), (3,4), (5,6), (6,7)\} \).

• An input refers to the first element in each pair (1, 3, 5, 6).
• An output refers to the second element in each pair (2, 4, 6, 7).

Each input must map, or correspond, to one and only one output. This rule sustains the focus on a one-to-one mapping, where no outputs are "shared" between different inputs. Function mapping is straightforward as long as we maintain this unique pairing.

Inverse Function

An inverse function is essentially the reverse of the original function. If a function is described as \( f(x) = y \), its inverse function \( f^{-1}(y) = x \) swaps the roles of inputs and outputs. For a function to possess an inverse, it must be one-to-one.

In our case, since each input in the function set \( \{(1,2), (3,4), (5,6), (6,7)\} \) has a unique output, the function is one-to-one, and therefore, it has an inverse. Constructing the inverse involves swapping each pair in the function:

• The pair \((1,2)\) becomes \((2,1)\).
• Further pairs are swapped, resulting in \((4,3), (6,5), (7,6)\).

Thus, the inverse function \( r^{-1} \) is \( \{(2,1), (4,3), (6,5), (7,6)\} \).
This new set illustrates how we can determine what input yields a given output from the original function, just by switching elements.

Input-Output Relationship

The input-output relationship is a basic yet essential aspect of any function. It outlines how inputs are related to outputs in a systematic manner. In mathematical terms, this relationship demonstrates how we transition from one value (input) to another value (output).
In the provided exercise, the function \( r = \{(1,2), (3,4), (5,6), (6,7)\} \) illustrates this relationship clearly. Each input (1, 3, 5, 6) is paired with a distinct output (2, 4, 6, 7).

• For example, if we input 1 into the function, the output is 2.
• Similarly, inputting 3 gives us an output of 4.

The clarity and uniqueness in this relationship not only help identify a one-to-one function but also enable us to determine the potential for an inverse. Understanding the intricacies of how each output directly correlates with a specific input is crucial for grasping both the operation of functions and the concept of their inverses.