Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(1,6),(2,6),(3,6)\\}$$

Step-by-Step Solution

Answer

Domain: \( \{1, 2, 3\} \); Range: \( \{6\} \); It is a function.

Step 1: Identify the domain

The domain of a relation is the set of all first elements from each ordered pair. Looking at the relation \( \{(1,6),(2,6),(3,6)\} \), the first elements (or x-values) are 1, 2, and 3. Thus, the domain is \( D = \{1, 2, 3\} \).

Step 2: Identify the range

The range is the set of all second elements from the ordered pairs. In this case, the second element (or y-value) is 6 for all pairs. Hence, the range is \( R = \{6\} \).

Step 3: Determine if the relation is a function

A relation is a function if each element in the domain maps to exactly one element in the range. Here, every element from the domain \( \{1, 2, 3\} \) maps to the same element, 6, in the range \( \{6\} \). Therefore, this relation is a function.

Key Concepts

Domain and Range

Understanding the concepts of domain and range is essential when dealing with relations and functions. The domain of a relation refers to the complete set of all possible input values, primarily represented by the x-values in an ordered pair. Essentially, it answers the question: 'What can we plug into this function?' In our example, looking at the relation \((1,6), (2,6), (3,6)\), we identify that the first numbers of these ordered pairs are 1, 2, and 3. Hence, the domain is \(D = \{1, 2, 3\}\).
The range, on the other hand, is the set of all possible output values, represented by the y-values in the pairs. In simpler terms, it tells us what outputs the function can achieve given the inputs from the domain. For the provided relation, the second numbers in each pair are all 6. So, the range is \(R = \{6\}\). This means that no matter what input we select from the domain, the output is always 6.

Ordered Pairs

Ordered pairs are fundamental in representing relations and functions. An ordered pair, presented as \((x, y)\), contains two elements: the first element (x) represents an input, and the second element (y) represents an output. The structure signifies a direct correlation between an input and its corresponding output.
For our relation \(\{(1,6), (2,6), (3,6)\}\), each pair ties a domain member with a range member. This specific schema allows us to easily track dependencies and associations between variables in a relation. It's like mapping the path from a number in the domain to its outcome in the range. Understanding ordered pairs helps solidify the foundation of comprehending more complex relationships and function behaviors.

Function Definition

Defining a function requires understanding the special relationship between sets of inputs and outputs. A relation is regarded as a function if every input (from the domain) pairs with exactly one output (from the range). This characteristic ensures uniformity in predictions and eliminates ambiguity.
Let's take our example \(\{(1,6), (2,6), (3,6)\}\). Here, each x-value (1, 2, or 3) is uniquely associated with the same y-value of 6. Despite the repetition in outputs, this unambiguous assignment corroborates the criteria for a relation to qualify as a function. Thus, a function guarantees that for every x, there will always be one and only one y value, ensuring reliability in mapping domain elements to range elements. This systematic predictability is what differentiates functions from simple relations.