# Multi-value Function: What & Why

• MHB
• highmath
In summary, a multi-valued function is considered a function because it meets the fundamental definition of a function, even though it may have multiple outputs for a single input. They are used in various mathematical and scientific applications and have been extensively studied and proven to be valid functions."
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Why multi-valued function is a function?

You can always make a multi-valued function $f$ between sets $X$ and $Y$ single-valued by considering the associated function (in the narrow, traditional sense) that maps $X$ to the power set of $Y$.

In the context of set-valued analysis (which has many applications in e.g. microeconomics), the multi-valued functions are often called "correspondences". Some problems from the application domain can then be translated elegantly into questions about those correspondences. If you are interested, I can provide more references.

In other contexts, such as complex analysis, multi-valued functions often arise as inverses, and then one typically make a choice by convention and calls it the "principal value".

A multi-valued function is still considered a function because it follows the fundamental definition of a function, which states that for every input, there is a unique output. Even though a multi-valued function may have more than one output for a single input, it still satisfies this requirement. Additionally, multi-valued functions are used in many mathematical and scientific applications, and they have been studied and proven to have important properties and characteristics. Therefore, they are considered valid and important functions in their own right.

## What is a Multi-value Function?

A multi-value function is a mathematical concept that assigns more than one output value to a single input value. This is in contrast to a traditional function, which only has one output value for every input value.

## Why are Multi-value Functions used?

Multi-value functions are used to model real-world situations where a single input can have multiple outcomes. They are also useful in solving certain mathematical problems, such as finding the roots of equations.

## What are some examples of Multi-value Functions?

Examples of multi-value functions include the inverse trigonometric functions (arcsin, arccos, arctan), which have multiple possible output values for a given input value. Another example is the square root function, which can have both a positive and a negative output for a given input.

## How are Multi-value Functions represented?

Multi-value functions are typically represented using a graph, with the input values on the x-axis and the output values on the y-axis. In some cases, a multi-value function may also be represented using a table of values.

## What is the difference between a Multi-value Function and a Relation?

A multi-value function is a type of relation, but not all relations are multi-value functions. A relation is a set of ordered pairs, while a multi-value function is a relation that follows specific rules, such as each input having multiple corresponding outputs. In other words, a multi-value function is a specific type of relation that has multiple outputs for a single input.

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