Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The set of ordered pairs \(\\{(-8,-2),(-6,0),(-4,0) (-2,2),(0,4),(2,-2)\\}\) represents a function.

Step-by-Step Solution

Answer

The set of ordered pairs \((-8,-2),(-6,0),(-4,0), (-2,2),(0,4),(2,-2)\) represents a function.

Step 1: Identify the x-values

The first component in each pair is the x-value. For the given set, they are -8, -6, -4, -2, 0, 2. Note that none of these values appear more than once.

Step 2: Verify unique y-values for each x-value

The second component in each pair is the y-value corresponding to the x-value. For the provided set of ordered pairs, each x-value has a unique y-value. That is, -8 is paired with -2, -6 with 0, -4 with 0, -2 with 2, 0 with 4 and 2 with -2. None of the y-values is linked to an already used x-value.

Step 3: Conclude on whether the set represents a function

Since each x-value in the set of ordered pairs is paired with a unique y-value, by the definition of a function, this implies that the given collection of ordered pairs represents a function.

Key Concepts

Function Definition

Understanding functions is integral to grasping mathematics. By definition, a function is a relation between two sets that assigns to each element of the first set, commonly called the domain, exactly one element of the second set, known as the range.

When visualizing this in terms of ordered pairs, which consist of an x-value (from the domain) and a y-value (from the range), a function would mean that for every x-value, there must be one and only one y-value associated with it. This connection is often described as a 'mapping' because we can visualize it as a clear pathway from an input (x) to an output (y).

If you encounter any set of ordered pairs, such as \(\{(-8,-2),(-6,0),(-4,0),(-2,2),(0,4),(2,-2)\}\), determining if it represents a function involves checking this principle of unique mapping.

X-value and Y-value Relations

In mathematics, the concept of order is pivotal. For ordered pairs, the x-value precedes the y-value, forming an (x, y) relationship. Each x-value in a function must relate to a single y-value, though a y-value may be associated with multiple x-values.

This hierarchy in ordered pairs ensures that each input (x-value) has a distinct output (y-value), which is how functions maintain their definition rigorously. For example, if you're given the set \(\{(-8,-2),(-6,0),(-4,0),(-2,2),(0,4),(2,-2)\}\), determining its function status requires you to scrutinize these relationships, ensuring that each x-value is a unique member of the domain leading to a specific y-value in the range.

Uniqueness of a Function

The uniqueness of a function is an essential criterion that distinguishes it from a non-function relation. Each x-value, serving as an input, should map to exactly one y-value, an output - never to multiple y-values.

This aspect of uniqueness is what makes a function predictable and reliable in terms of understanding the relationship between variables. As clarified in our exercise, the list of ordered pairs \(\{(-8,-2),(-6,0),(-4,0),(-2,2),(0,4),(2,-2)\}\) exhibits this uniqueness, with every x-value corresponding to exactly one y-value.

Identifying Functionality

When looking at the set of ordered pairs, notice how no x-value is repeated with a differing y-value. This confirms that for that particular collection, it does symbolize a function as per the strict definition used in mathematics.