Problem 9 - Algebra A Combined Function

Question

Find the domain and the range of each relation. Also determine whether the relation is a function. $$ \\{(-3,-3),(0,0),(3,3)\\} $$

Step-by-Step Solution

Answer

Domain: \{-3, 0, 3\}, Range: \{-3, 0, 3\}, Function: Yes.

Step 1: Define the Domain

The domain of a relation is the set of all first elements (x-values) in the ordered pairs. For the given relation \( \{(-3,-3),(0,0),(3,3)\} \), identify and list all x-values. These are -3, 0, and 3. Therefore, the domain is \( \{-3, 0, 3\} \).

Step 2: Define the Range

The range of a relation is the set of all second elements (y-values) in the ordered pairs. For the given relation \( \{(-3,-3),(0,0),(3,3)\} \), identify and list all y-values. These are -3, 0, and 3. Therefore, the range is \( \{-3, 0, 3\} \).

Step 3: Determine if the Relation is a Function

A relation is a function if each element in the domain is associated with exactly one element in the range. In the given relation \( \{(-3,-3),(0,0),(3,3)\} \), each x-value -3, 0, and 3 is paired with one unique y-value -3, 0, and 3 respectively. Hence, the relation is a function.

Key Concepts

Relations and Functions

Understanding the difference between relations and functions is one of the fundamental concepts in mathematics. A "relation" is simply a set of ordered pairs. In these pairs, each element from one set is related to an element from another set. It's similar to a "matching" or "pairing" concept.
In contrast, a "function" is a special type of relation. Here, each input (usually the x-value) is related to exactly one output (the y-value). This means, for a relation to be a function, no two ordered pairs should have the same first element with different second elements.
To summarize, while all functions are relations, not all relations are functions. An easy way to identify a function is by using the "vertical line test" on a graph representing the relation. If a vertical line intersects the graph more than once, then the relation is not a function.

Ordered Pairs

An ordered pair is a fundamental part of coordinate geometry and algebra, consisting of two elements organized in a specific order, usually written as \((x, y)\). Here, \(x\) is the first component (or abscissa), and \(y\) is the second component (or ordinate).
Ordered pairs are used to describe the location of a point on a coordinate plane, where \(x\) refers to the horizontal position and \(y\) refers to the vertical position. This ordering is critical because \((x, y)\) is different from \((y, x)\).

• The first element is often referred to as "input," "independent variable," or "domain value."
• The second element is called "output," "dependent variable," or "range value."

By analyzing ordered pairs, one can determine relationships and map changes from one set of numbers to another, especially in real-world applications, where they indicate phenomena like time versus distance or effort versus result.

Function Definition

A function is a specific type of relation defined by the "input-function output" rule. In simple terms, for each input, there is a unique output. Function definition helps in understanding complex data relationships and how one quantity affects another.
Every function consists of a domain (inputs) and a range (outputs). The function map associates each input from the domain to precisely one output in the range. This is often expressed as \(f(x) = y\), where \(x\) is the input and \(y\) is the output.
One way to check if a relation is a function is to ensure that every \(x\)-value (or domain element) has one and only one \(y\)-value related to it. In our case, the relation \(\{(-3,-3),(0,0),(3,3)\}\) is a function because no \(x\)-value repeats with a different \(y\)-value. Functions are powerful tools in math, allowing us to predict unknown data and solve real-world problems by establishing consistent rules.