Problem 47 - Algebra 2

Question

REVIEW Which set of points describes a function? $$ \begin{array}{l}{\mathbf{F}\\{(3,0),(-2,5),(2,-1),(2,9)\\}} \\\ {\mathbf{G}\\{(-3,5),(-2,3),(-1,5),(0,7)\\}} \\\ {\mathbf{H}\\{(2,5),(2,4),(2,3),(2,2)\\}} \\\ {\mathbf{J}\\{(3,1),(-3,2),(3,3),(-3,4)\\}}\end{array} $$

Step-by-Step Solution

Answer

Set G describes a function, as each x-value has a unique y-value.

Step 1: Understand the Definition of a Function

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In terms of coordinate points, if any x-value repeats but has a different y-value for those points, then the set is not a function.

Step 2: Analyze Set F

In set F, we have the points \((3,0),(-2,5),(2,-1),(2,9)\). Here, the x-value '2' repeats but has different y-values (-1 and 9). Therefore, set F does not describe a function.

Step 3: Analyze Set G

In set G, the points are \((-3,5),(-2,3),(-1,5),(0,7)\). No x-values are repeating. Each x-value \(-3, -2, -1, 0\) is associated with exactly one unique y-value. Therefore, set G describes a function.

Step 4: Analyze Set H

Set H includes the points \((2,5),(2,4),(2,3),(2,2)\). Here, the x-value '2' repeats multiple times and is associated with different y-values (5, 4, 3, and 2). Thus, set H does not describe a function.

Step 5: Analyze Set J

In set J, the points are \((3,1),(-3,2),(3,3),(-3,4)\). The x-values '3' and '-3' repeat with different associated y-values (1 and 3 for 3; 2 and 4 for -3). Therefore, set J does not describe a function.

Key Concepts

Coordinate Points

In mathematics, coordinate points are used to represent locations on a plane. Each point is defined by a pair of numbers \(x, y\), where \(x\) is the horizontal component and \(y\) is the vertical component. These points tell us exactly where on the grid a specific location can be found.
Coordinate points are essential in defining functions, especially when examining relationships between inputs and outputs. It is important to note how coordinates share relations in various mathematical contexts, especially when discussing whether a set of coordinates forms a function.

Input-Output Relation

A function represents a specific kind of input-output relation. To be a function, every input must have exactly one output. This can be visualized using coordinate points, where each \(x\)-value (input) must map to only one \(y\)-value (output).
Understanding how inputs and outputs are related helps you determine if you have a function. If you find that a single input gives multiple outputs, the relation is not a function. Keeping this in mind ensures that the mapping stays logical and consistent.

Unique Y-Values

In the context of functions, unique \(y\)-values play a crucial role in determining if a relation is a valid function. Simply put, each \(x\)-value in your coordinate points should pair with exactly one \(y\)-value.
For example, in a valid function, if \(x = 2\) is repeated, it should not have different associated \(y\)-values. If it does, this violates the function rule of having unique \(y\)-values for each input. Checking the uniqueness of \(y\)-values is a quick way to verify the function property of coordinate points.

X-Value Repetition

When evaluating if a set of coordinate points describes a function, the repetition of \(x\)-values is a major factor. If an \(x\)-value appears more than once but corresponds to different \(y\)-values, the set does not form a function.
Repetition of \(x\)-values with different outputs breaks the core rule of what a function is: each \(x\) with its unique \(y\). This principle helps prevent confusion in identifying valid functions and aids in distinguishing other types of relations.