Problem 47 - Introductory Algebra for College Students

Question

If a relation is represented by a set of ordered pairs, explain how to determine whether the relation is a function.

Step-by-Step Solution

Answer

A relation represented by a set of ordered pairs is a function if each first element of the ordered pairs is associated with exactly one second element. This means that within the set of ordered pairs, there must not be any two pairs with the same first element and different second elements.

Step 1: Understand what a function is

A function is a relationship between two sets that associates each element of the first set (the domain) to exactly one element of the second set (the range). Within the context of ordered pairs, this translates to the rule that each first element of an ordered pair is associated with exactly one second element. In other words, in the set of ordered pairs, there cannot be two pairs \((x, y_1)\) and \((x, y_2)\) such that \(y_1\) does not equal \(y_2\).

Step 2: Analyze the Set of Ordered Pairs

Look closely at the set of ordered pairs that represent the relation. Check each pair and make note of the first elements. If any first element appears in more than one ordered pair with a different second element, then the relation is not a function.

Step 3: Make a Conclusion

After analyzing the set of ordered pairs, if no first elements repeat with different second elements, then the relation is a function. Otherwise, the relation is not a function.

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is simply a collection of two elements arranged in a specific sequence. You'll often encounter them in the form \((x, y)\), where \(x\) is the first element and \(y\) is the second. This is different from a basic set because the order matters — \((x, y)\) is different from \((y, x)\).

Ordered pairs are crucial because they allow us to track and map relationships between two sets of data. When dealing with a list of ordered pairs, each individual pair represents a precise point of information, often visualized on a graph as a dot where the first value (called the abscissa or \(x\)-value) represents the horizontal position and the second value (called the ordinate or \(y\)-value) represents the vertical one.

• The first item in the pair is often considered as input.
• The second item is usually deemed output.

Understanding ordered pairs is foundational in delving into concepts like relations and functions, as each pair serves as a building block for these more complex ideas.

Domain and Range

When we discuss functions, two key concepts emerge: the domain and the range. The domain is essentially the set of all potential "input" values (first elements of ordered pairs), while the range includes all possible "output" values (second elements of ordered pairs).

Think of the domain as the collection of all starting points in the ordered pairs, while the range consists of the destinations these points reach.

• The domain is composed of all first elements in any relation.
• The range consists of all second elements in that relation.

For example, in the set of ordered pairs \(((1, 2), (3, 4), (5, 6))\), the domain would be \(\{1, 3, 5\}\), while the range would be \(\{2, 4, 6\}\). This understanding helps you determine whether a relation can be considered a function.

Relation vs Function

To understand the difference between a relation and a function, think of a relation as any connection between elements from two sets. A function is a specific type of relation with a unique rule: each element in the domain (first value of the ordered pair) must map to exactly one element in the range (second value of the ordered pair).

In simpler terms, for something to be a function, you cannot have two ordered pairs like \( (x, y_1) \) and \( (x, y_2) \) if \( y_1 eq y_2 \). It's sort of like saying that in a function, each input has only one output.

• A relation can have many outputs for a single input.
• A function ensures only one output for each input.

So, if you're examining a set of ordered pairs and you find any instances where the first value repeats with different second values, it's not a function, but merely a relation. This distinction is important as it sets the rules for how we process and interpret data in mathematics.