91Ó°ÊÓ

In mathematics, every function has two important parts: the domain and the range.
The domain of a function is the complete set of possible values of the independent variable, usually represented by the letter \( x \). This means, basically, all the \( x \)-values you can plug into the function to get a \( y \)-value back. Conversely, the range is the set of all possible \( y \)-values which we will get after plugging in the \( x \)-values into the function.

For a relation to be considered a function, each \( x \)-value in the domain must relate to exactly one \( y \)-value in the range.

• If any \( x \)-value is paired with more than one \( y \)-value, then we do not have a function.

In our original problem, the presence of multiple \( y \)-values for a given \( x \), such as both 4 and -4 for \( x = 3 \), prevents the relation from being a function.

Ordered pairs are a fundamental concept when dealing with relations and functions.
They're simply pairs of numbers, usually written in the form \((x, y)\), with \( x \) representing a point on the horizontal axis, and \( y \) representing a point on the vertical axis.

These pairs describe a specific relationship between \( x \) and \( y \), and are used to map the connection between inputs and outputs. Each pair tells you that for a certain input \( x \), we receive a specific output \( y \).

• In the context of our problem, we look at ordered pairs such as \((3, 4)\) and \((3, -4)\) to determine if the relation constitutes a function.

If any \( x \)-value appears more than once with different \( y \)-values, like \( (3, 4) \) and \( (3, -4) \), the relation does not satisfy the criteria of defining a function.

To understand functions, we must first grasp the precise definition.
A function in mathematics is defined as a special relation between sets. Here, every element in the first set (the domain) is associated with exactly one element in the second set (the range).

This means each \( x \)-value should connect to a single \( y \)-value, ensuring clarity in mapping inputs to outputs without ambiguity.

• In our example exercise, multiple \( y \)-values (e.g., 4 and -4) linked to the same \( x \)-values (e.g., \( x = 3 \)) breach this rule.

These violations in what the function definition stipulates demonstrate why such a relation doesn't constitute a proper function. Understanding this pivotal concept helps avoid confusion when examining relations.