Problem 12 - Algebra 2 and Trigonometry

Question

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(1,4),(2,7),(1,10),(4,13)\\} $$

Step-by-Step Solution

Answer

There is no inverse function because the input "1" maps to two different outputs.

Step 1: Understand the Definition

A function has an inverse if it is a one-to-one function. This means each output is linked to exactly one input without repeats. Check each pair to ensure each input maps to a unique output.

Step 2: Check for Unique Inputs

Examine the set \(\{(1,4),(2,7),(1,10),(4,13)\}\). Notice that the input "1" is mapped to both "4" and "10". This indicates that the same input maps to more than one output, violating the one-to-one function requirement.

Step 3: Determine If Function Has Inverse

Since the function maps the input "1" to two different outputs, it is not a one-to-one function. As a result, this function does not have an inverse.

Key Concepts

Understanding One-to-One Functions

A one-to-one function is a crucial concept in determining if a function has an inverse. For a function to be one-to-one, each output in the set must be paired with a unique input. This means that no two different inputs will produce the same output. In simpler terms, every single input has its own unique output, and vice versa.
Therefore, if you find any repeated inputs leading to different outputs or any outputs being produced by multiple different inputs, the function would not be classified as one-to-one. This is essential because only one-to-one functions have inverses, where each input can be traced back to exactly one output. Understanding this allows you to check if a function has the potential to be inverted.

Diving into Function Mapping

Function mapping describes how each input of a function is connected to an output. Consider it like a road map where each road (input) leads to a specific destination (output). In a one-to-one function, every road heads exclusively to a different destination. This mapping ensures that as you trace the road, you end up at a unique place.
For the given set of pairs \(\{(1,4),(2,7),(1,10),(4,13)\}\), function mapping shows where each input number connects to an output number. When examining function mapping, we ensure that no single piece of the map covers more than one route, which would result in too many paths leading to a particular destination.
In cases like our exercise, where input "1" goes to both "4" and "10", the mapping does not abide by one-to-one rules, indicating potential issues with forming an inverse from the function.

Exploring Unique Input-Output Pairs

Unique input-output pairs are fundamental when confirming if a function can have an inverse. In any given function, this uniqueness signifies that each pair consists of an input directly connected with only one specific output. Thus, each couple or pair should be able to represent one unbroken relationship.

• A pair like \((1,4)\) suggests that the input "1" gives the output "4".
• The pair \((2,7)\) indicates that "2" results in "7".

When pairs repeat inputs or mismatch outputs, like \((1,10)\) happening alongside \((1,4)\), it breaks the rule of one-to-one uniqueness required for an inverse. The redundancy or overlapping inputs in these pairs highlights that the function provided isn't carefully organized with each member having distinct correspondence, reflecting its incapability to form an inverse successfully.