91Ó°ÊÓ

In mathematics, an ordered pair is a set of two numbers used to represent a point on a graph, typically in a two-dimensional coordinate system. They are written in the form \(x, y\), where \(x\) is the first element of the pair, and \(y\) is the second element. The order of these numbers is crucial because \( (x, y) \) is different from \( (y, x) \).
Ordered pairs are a fundamental concept in algebra as they allow us to plot and understand relationships between variables:

• The first number in the pair, \(x\), represents the horizontal position, or the axis commonly referred to as the "x-axis." This is why \(x\) values are often called the inputs or independent variables.
• The second number, \(y\), represents the vertical position, or the "y-axis," known as the output or dependent variable.

In the exercise, for each \(x\) in the set \{-3, 0, 3\}, the \(y\) value remains constant as \(-4\), creating ordered pairs such as \((-3, -4)\), \(0, -4)\), and \(3, -4)\). This was derived given that the function \(y = -4\) applies irrespective of the change in \(x\) values.

A constant function is a type of linear equation where the output value, usually denoted as \(y\), remains unchanged, no matter what the input value, or \(x\), might be.
The general form of a constant function is \(y = c\), where \(c\) is a constant value. Here, the function \(y = -4\) indicates that \(y\) will always equal \(-4\) for any \(x\).
This type of function is represented graphically as a horizontal line:

• Since the value of \(y\) does not change, this implies that every point on the line has the same \(y\) value.
• No matter how \(x\) varies, from negative to positive, the \(y\) value remains constant.

Understanding a constant function helps students recognize how certain changes (or lack of them) in \(y\), unaffected by \(x\), can lead to simple yet powerful expressions.

The set of \(x\)-values in a problem provides the possible inputs into a function or equation. In the presented exercise, \{-3, 0, 3\} represents the specific set of \(x\) values to be used. Each of these values is substituted into the function to calculate a corresponding output or \(y\) value.
With this set, we:

• Start by substituting each \(x\) into the function \(y = -4\).
• See that no matter what \(x\) is, the function always returns the same \(y\) value of \(-4\).

While this might seem straightforward for a constant function, recognizing the \(x\) values set is vital in more complex functions. It defines the domain of inputs over which any function will operate. In more diverse contexts, determining the appropriate \(x\) values can significantly affect the calculation of different outputs.