Problem 5 - Algebra A Combined Function

Question

Find the domain and the range of each relation. Also determine whether the relation is a function. $$ \\{(1,1),(1,2),(1,3),(1,4)\\} $$

Step-by-Step Solution

Answer

Domain: \(\{1\}\); Range: \(\{1, 2, 3, 4\}\); Not a function.

Step 1: Identify the Domain

The domain of a relation is the set of all the first elements in each ordered pair. Here, the relation is \(\{(1,1),(1,2),(1,3),(1,4)\}\). For each pair, the first element is just 1. Therefore, the domain is \(\{1\}\).

Step 2: Identify the Range

The range of a relation is the set of all the second elements in each ordered pair. Observing the second elements in each pair, we have 1, 2, 3, and 4. Therefore, the range is \(\{1, 2, 3, 4\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if each element in the domain is paired with exactly one element in the range. Here, the single domain element 1 is paired with multiple elements (1, 2, 3, and 4), which means there are multiple outputs for a single input. Thus, this relation is not a function.

Key Concepts

Domain

In mathematics, the domain of a function or relation is a critical concept, especially when dealing with ordered pairs. The domain is defined as the set of all possible inputs or the first elements in each ordered pair of a relation. These inputs determine what we feed into the relation to get an output.
For instance, consider the relation \(\{(1,1),(1,2),(1,3),(1,4)\}\). Here, each ordered pair consists of two numbers, where the first number is the input, known as the domain element. In this specific relation:

• The domain is simply \(\{1\}\), as 1 is the first number in each pair.

Understanding the domain helps us recognize the scope and limitations of a relation or a function. It tells us which inputs are applicable. In this example, because all ordered pairs share the same first element, our domain becomes very limited. It's just \(\{1\}\). This simplicity in the domain is often a telltale sign that the relation might not be a function, as a function typically requires each input to map uniquely to one output.

Range

Now, let's talk about the range. The range of a relation involves all possible outputs, or the second elements in each of the ordered pairs. It is essential because it tells us the possible outcomes that result from using the domain elements.
For our relation \(\{(1,1),(1,2),(1,3),(1,4)\}\):

• The range comprises the second numbers: \(\{1, 2, 3, 4\}\).

This shows the possible values that are associated with the single input (domain) value of 1. The range lets us understand the output spread and variety we can expect from our relation. Knowing how sometimes range elements repeat or are unique across different relations can also hint at whether the relation might qualify as a function.

Ordered Pairs

Ordered pairs are fundamental in expressing relations and functions. They consist of two elements: the first representing an input or domain, and the second representing an output, or range. Ordered pairs are typically written in (x, y) format, where x is the first element, and y is the second element.
In the relation \(\{(1,1),(1,2),(1,3),(1,4)\}\) :

• The first number in each ordered pair is consistently 1, indicating a static domain value.
• The second number represents potential outcomes (1, 2, 3, 4).

Ordered pairs are critical for identifying both the domain and range. Examining these pairs closely helps us determine if each input (first elements) maps to a unique output (second elements). In this case, noticing that 1 maps to multiple values (1, 2, 3, 4) suggests that the relation does not meet the criteria of a function, which requires unique mappings for each domain element.