Problem 60 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the relation represents a function. \(\\{(-1,-1),(-2,-2),(-3,-3)\\}\)

Step-by-Step Solution

Answer

The relation is a function.

Step 1: Understand the Definition of a Function

A function is a relation where each input (or domain value) maps to exactly one output (or range value). This means each 'x' value in the set should be associated with only one 'y' value.

Step 2: Identify the Domain

Identify all the 'x' values in the relation, which are -1, -2, and -3. This is the domain of the relation.

Step 3: Identify the Range

Identify all the 'y' values in the relation, which are also -1, -2, and -3. This is the range of the relation.

Step 4: Check for Unique Mapping of Domain

Confirm that each 'x' value is paired with exactly one 'y' value. Here, -1 is paired with -1, -2 with -2, and -3 with -3.

Step 5: Determine if the Relation is a Function

Since each element of the domain is paired with exactly one element of the range, the relation represents a function.

Key Concepts

Domain and Range in Functions

When working with functions, two important concepts to understand are the domain and range. The domain is essentially the list of all possible inputs for the function, usually represented by 'x' values. In any given math problem like the one we're analyzing here, you need to identify every unique 'x' value in the set of ordered pairs. For the relation \(\{(-1,-1),(-2,-2),(-3,-3)\}\), the domain includes the elements -1, -2, and -3.
The range, on the other hand, is the set of all possible outputs or 'y' values. Just like determining the domain, you go through the set of ordered pairs and list out all unique 'y' values. In the case of this problem, the range is exactly the same as the domain, consisting of -1, -2, and -3.

The domain tells you what can go into the function.

The range tells you what can come out of the function.

By fully understanding these two aspects, you'll be much better equipped to handle any function-related problems you encounter.

Understanding Relations

In its simplest form, a relation is a connection between two sets, usually described by ordered pairs like (x, y). These pairs suggest a certain input yields a specific output. Graphically, this can be visualized as a set of points on a coordinate plane.
Not every relation qualifies as a function, but all functions are relations. The key difference is that functions have stricter rules: for each input, there can only be one output. This means if you had more than one 'y' value for a single 'x' value in your set, it wouldn't be a function.
In our example \(\{(-1,-1),(-2,-2),(-3,-3)\}\), each 'x' value is paired with one and only one 'y' value, maintaining a perfect one-to-one connection. This illustrates a functional relationship where inputs and outputs are neatly aligned.

The Concept of Unique Mapping

A crucial aspect of functions is unique mapping. This essentially means that each item in the domain must map to a single, distinct item in the range. Unique mapping is what separates a function from a simple relation.
For instance, if you assess the set \(\{(-1,-1),(-2,-2),(-3,-3)\}\), you'll notice:

• -1 is paired with -1
• -2 is paired with -2
• -3 is paired with -3

This example demonstrates unique mapping because no 'x' value shares a 'y' output with another. It's this kind of mapping that guarantees that the relation is indeed a function. As you grow more comfortable identifying these mappings, understanding and direct application of the function concept become more intuitive.