Problem 62 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the relation represents a function. \(\\{(2,5),(7,11),(15,8),(7,9)\\}\)

Step-by-Step Solution

Answer

This relation is not a function, as the x-value '7' has two different outputs.

Step 1: Understand the Definition

A relation is considered a function if each input (or domain value) is associated with exactly one output (or range value). In simpler terms, for a set of ordered pairs, a function has no repeated x-values.

Step 2: Identify the x-values

Look at the given relation \(\{(2,5),(7,11),(15,8),(7,9)\}\) and identify the set of x-values. These are \(\{2, 7, 15, 7\}\).

Step 3: Check for Repeated x-values

Check the identified x-values \(\{2, 7, 15, 7\}\). Notice that the x-value '7' appears twice (in pairs \((7,11)\) and \((7,9)\)). This repetition means there is more than one output for a single input.

Step 4: Conclude on the Functionality

Since the x-value '7' is associated with more than one y-value (both 11 and 9), the relation \(\{(2,5),(7,11),(15,8),(7,9)\}\) does not satisfy the criteria of a function, as it does not have exactly one output for each input.

Key Concepts

Relations

In algebra, a relation is essentially a set of ordered pairs. Think of it as a collection of couples, where each couple consists of a pair of values. The first value of each pair is called the "input," while the second one is the "output." Relations can associate multiple outputs with one input or have an individual output for each input.
For example, in the relation \(\{(2,5),(7,11),(15,8),(7,9)\}\), each pair comprises x-values and y-values. It's helpful to picture this in everyday contexts. Imagine, for instance, that each x-value represents a student's ID and each y-value represents their score in a test.
Not all relations are functions. In a function, every input is linked to exactly one output. This unique association is what distinguishes a function from a general relation.

Domain and Range

Two essential components of relations and functions are "domain" and "range." The domain refers to all possible input values, often represented as x-values. Meanwhile, the range encompasses all possible output values, represented as y-values. Suppose we have a relation like \(\{(2,5),(7,11),(15,8),(7,9)\}\). The x-values (2, 7, 15) form the domain, while y-values (5, 11, 8, 9) constitute the range.
In the context of identifying functions, a crucial step is examining the domain. Why? If any x-value repeats and is paired with different y-values, then the relation cannot be a function. That's because a single input corresponds to more than one output, violating the rule of a function.
Understanding domain and range doesn't just help in determining whether a relation is a function, but also sets the foundation for working with linear equations, graphs, and other algebraic concepts.

Ordered Pairs

Ordered pairs are the fundamental building blocks of relations in algebra. They consist of two elements, written in a specific sequence inside parentheses, like (x, y). The order matters because the first element, x, signifies the input or independent variable, and the second element, y, indicates the output or dependent variable.
When analyzing ordered pairs, such as in the relation \(\{(2,5),(7,11),(15,8),(7,9)\}\), it’s important to note how the relationship between x and y values functions. Ordered pairs not only provide a visual representation of data, but they also embody the input-output scheme that defines many mathematical operations and concepts.
Additionally, analyzing a set of ordered pairs can reveal patterns and relationships that aren't immediately obvious. In contexts like determining functions, identifying repeated x-values or checking the consistency of these pairs in representing a rule or pattern becomes crucial. This understanding lays the groundwork for graphing equations and exploring more complex algebraic ideas.