Problem 25 - A Graphical Approach to College Algebra

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. $$\\{(5,1),(3,2),(4,9),(7,6)\\}$$

Step-by-Step Solution

Answer

Domain \(D\) is \(\{3, 4, 5, 7\}\), Range \(R\) is \(\{1, 2, 6, 9\}\), and it is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all first values in the ordered pairs. For the set \((5,1),(3,2),(4,9),(7,6)\), the domain can be found by listing all the first elements of each pair: \(5, 3, 4, 7\). Thus, the domain \(D\) is \(\{3, 4, 5, 7\}\).

Step 2: Identify the Range

The range of a relation is the set of all second values in the ordered pairs. Looking at our set \((5,1),(3,2),(4,9),(7,6)\), the range can be identified by listing all second elements: \(1, 2, 9, 6\). Therefore, the range \(R\) is \(\{1, 2, 6, 9\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if each element in the domain corresponds to exactly one element in the range. In the given set \((5,1),(3,2),(4,9),(7,6)\), each domain element \(3, 4, 5, 7\) is associated with a unique range element \(2, 9, 1, 6\), respectively. Since no domain element is repeated with different range values, the relation is a function.

Key Concepts

Exploring Domains and Ranges

In mathematics, understanding the concepts of domain and range is essential when dealing with relations and functions. The **domain** is the set of all possible inputs or "first values" in a relation. For instance, given the set of ordered pairs \((5,1),(3,2),(4,9),(7,6)\), the domain would consist of \[5, 3, 4, 7\]. These are the 'x' values or the first component of each ordered pair.
On the other hand, the **range** encompasses all possible outputs or "second values" in the relation. Using the same set, the range includes \[1, 2, 9, 6\], which are the 'y' values or the second component of each ordered pair.
Understanding these two sets is vital:

• The domain tells us which values we can use as input in our relation or function.
• The range tells us the possible outcomes for these inputs.

By identifying these sets, we can better grasp the behavior of relations and functions.

Understanding Functions

A **function** is a special type of relation in mathematics. For a relation to be classified as a function, each element in the domain must be associated with exactly one element in the range.
Let's consider our set of ordered pairs: \((5,1),(3,2),(4,9),(7,6)\). Each domain value \(3, 4, 5, 7\) is linked to exactly one range value \(2, 9, 1, 6\), respectively.
Here's the key characteristic of a function:

• No two ordered pairs can have the same first element with different second elements.

In simple terms, if you think of a function like a vending machine, every button (domain value) brings you exactly one kind of snack (range value). Some buttons could lead to the same snack, but no button triggers two different snacks.
By ensuring this, we can predictably map each input to a single outcome.

Decoding Ordered Pairs

In both relations and functions, **ordered pairs** are a foundational concept. An ordered pair \((x, y)\) consists of two elements where the order matters substantially. The first element \(x\) is from the domain, and the second element \(y\) is from the range.
For example, looking at \((5,1)\) from our set \((5,1),(3,2),(4,9),(7,6)\),

• "5" is the input or domain element, and
• "1" is the output or range element.

Ordered pairs help us precisely map inputs to outputs, essential for defining both relations and functions.
Remember, flipping the pair like \((1, 5)\) changes the meaning entirely, highlighting the importance of their order. Understanding and working with ordered pairs allows students to navigate more complex mathematical concepts with clarity.