Problem 36 - Algebra 2

Question

Find the domain and range of each relation and determine whether it is a function. $$ \\{(2,4),(4,8),(8,16)\\} $$

Step-by-Step Solution

Answer

The domain is {2, 4, 8}, the range is {4, 8, 16}, and it is a function because each domain element maps to a unique range element.

Step 1: Identify the Domain

The domain of a relation is the set of all first elements of the ordered pairs. In this case, look at the first number in each pair to determine the domain.

Step 2: Identify the Range

The range of a relation is the set of all second elements of the ordered pairs. In this case, look at the second number in each pair to determine the range.

Step 3: Determine if the Relation is a Function

To determine if the relation is a function, verify that each element of the domain maps to exactly one element of the range. There can be no duplicate first elements in the ordered pairs.

Key Concepts

Domain of a Relation

The domain of a relation is a fundamental concept in mathematics, particularly in the study of functions. It comprises all the possible input values that can be used in a set of ordered pairs. In simpler terms, the domain is the set of all the first components from each ordered pair in a relation.

For example, consider the set \( \{ (2,4), (4,8), (8,16) \} \). When we extract the first numbers of these pairs, we get the domain, which in this case is \( \{2, 4, 8\} \). It is crucial to thoroughly examine each first component, ensuring that no value is overlooked, which can be especially important for relations represented by a rule or formula.

Range of a Relation

The range of a relation represents all the conceivable output values that result from the input values within the domain. It consists of the second elements from each ordered pair in a relation. If the domain is like the ingredients you can use in a recipe, then the range is like the list of dishes you can cook with those ingredients.

As an example, taking our aforementioned set \( \{ (2,4), (4,8), (8,16) \} \), the range is the set that includes 4, 8, and 16, hence \( \{4, 8, 16\} \). Identifying the range involves focusing on the second number of each ordered pair. Remember that while some elements might be repeated in the range, each occurrence is derived from a unique ordered pair.

Ordered Pairs

Ordered pairs are the backbone of relations and functions within mathematics. These pairs consist of two elements where the order in which they are written is significant. The first number is the input, and the second number is the output. Notation for an ordered pair uses parentheses, separated by a comma, like (input, output).

Looking at our example \( \{ (2,4), (4,8), (8,16) \} \), each set of parentheses contains an ordered pair. The powerful aspect of ordered pairs is their ability to represent a relationship between two quantities in a clear and organized manner. When interpreting ordered pairs, it's essential to maintain their sequence since altering the order would potentially change the relationship they represent.

Function Definition

A function is a special type of relation that links each element of a domain to exactly one element in the range. The pristine characteristic of a function is that an input value must lead to only one output value, ensuring a one-to-one correspondence between values. Functions can be represented by equations, graphs, or sets of ordered pairs.

In evaluating whether a relation such as \( \{ (2,4), (4,8), (8,16) \} \) is a function, we ascertain that each domain element (2, 4, and 8) maps to a unique range element. Since there are no duplicate first elements in our ordered pairs and each input has a distinct output, we can affirm that the given relation is indeed a function.