Problem 63 - Algebra 2

Question

Which relation is not a function? F. \(\\{(0,9),(2,3),(3,2),(4,1)\\}\) H. \(\\{(0,3),(2,3),(3,3),(4,3)\\}\) G. \(\\{(3,2),(4,1),(0,9),(3,3)\\}\) J. \(\\{(0,3),(3,2),(2,4),(4,6)\\}\)

Step-by-Step Solution

Answer

The relation G is not a function because it maps the input 3 to two different outputs (2 and 3).

Step 1: Understanding the Criteria for a Function

To determine if a relation is a function, each input (commonly represented by the first component of the ordered pairs) must have exactly one output (the second component). This means that for any x-value, there must be only one corresponding y-value associated with it in the set of ordered pairs.

Step 2: Reviewing Each Relation

Examine each relation to see if any x-value (first number in the ordered pairs) is associated with more than one y-value (second number in the ordered pairs). F. No x-value is repeated, which means each x has exactly one y. H. No x-value is repeated, which means it passes the function test. J. No x-value is repeated, this is also a function. However, G has the x-value 3 associated with two different y-values (2 and 3), which violates the criteria for a function.

Step 3: Identifying the Non-Function

Relation G has a repeated x-value with different corresponding y-values, specifically the ordered pairs (3,2) and (3,3). Since the definition of a function requires each input to have exactly one output, and relation G does not meet this requirement, it is not a function.

Key Concepts

Functions and Relations

When studying algebra, understanding the difference between functions and relations is essential. Relations are any set of ordered pairs. For instance, say we have a collection of people and their favorite fruit, the pairing of each person to a unique fruit can be seen as a relation. However, a function is a special type of relation where every input (or domain element) is paired with exactly one output (or range element).

Imagine if some people have more than one favorite fruit; this scenario would still be a relation but not a function. Drawing from our original problem, we looked at sets of ordered pairs, inspecting whether they adhered to function criteria. Where a relation like H has all inputs producing the same output, it remains a function because each input still maps to one and only one output.

Ordered Pairs

In algebra, an ordered pair is a pair of elements written in a specific order - generally, as \( (x, y) \). The first component \( x \) corresponds to the input value, while the second component \( y \) corresponds to the output. Ordered pairs are often used to represent a relationship between two things, like the coordinates on a graph.

Ordered pairs that make up functions are special because they must pass the 'vertical line test' on a graph, meaning that a vertical line will only touch the graph at one point for any given \( x \) value. This aligns with the function criterion, ensuring every \( x \) is paired with a single \( y \) value, as seen in sets F, H, and J in the exercise.

Function Criteria

The critical condition for a set of ordered pairs to be deemed a function is straightforward: every \( x \) value must have one and only one \( y \) value. This requirement is called the function criteria. Sets that violate this rule, like set G in the exercise where the \( x \) value of 3 is paired with two different \( y \) values (2 and 3), are not functions.

To determine function eligibility, one must survey the set for repeated \( x \) values with different \( y \) counterparts. If you find even one instance of such a repetition, the set is not a function. This method was applied in Step 2 of the provided solution to weed out the non-function relations.

Input-Output Relationship

The core idea of a function in algebra revolves around the input-output relationship. Each input (or \( x \) value) in a function must be associated with exactly one output (or \( y \) value). In other words, a function assigns exactly one output to each input, creating a unique pairing that maps elements of the domain (inputs) to elements of the range (outputs).

If you consider a vending machine as an analogy, inserting a specific amount of currency (input) will get you only one selected snack (output). Similarly, for the sets F, H, and J, we observed that each input led to a single, predictable output, maintaining the purity of the function relationship.