Problem 7 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ [(-3,-3),(-2,-2),(-1,-1),(0,0)] $$

Step-by-Step Solution

Answer

The given relation is a function. The domain of the function is {-3, -2, -1, 0}, and the range is {-3, -2, -1, 0}.

Step 1: Checking if the Relation is a Function

Check whether each element from the domain (-3, -2, -1, 0) maps to exactly one element in the range (-3, -2, -1, 0). In this case, each element indeed maps to exactly one element. Hence, the given relation is a function.

Step 2: Determining the Domain of the Function

The domain of a function is the set of all first coordinates (x-values) from the ordered pairs. In this case, the domain is the set of all x-coordinates, which are -3, -2, -1 and 0.

Step 3: Determining the Range of the Function

The range of a function is the set of all second coordinates (y-values) from the ordered pairs. In this case, the range is the set of all y-coordinates, which are -3, -2, -1 and 0.

Key Concepts

Domain and Range

In algebra, understanding the domain and range is key to analyzing functions. The **domain** of a function is the set of all possible input values or x-values. These are the values you can plug into a function.
For example, in the ordered pairs \([(-3,-3),(-2,-2),(-1,-1),(0,0)]\), the domain is the first elements: -3, -2, -1, and 0.
On the other hand, the **range** is the set of all possible output values or y-values. It represents what you get after applying the function to all domain values.
In our example, the range includes -3, -2, -1, and 0, as these are the second elements in each pair.
It’s crucial to list these values without repetition, setting the foundation for analyzing functions.

Relations and Functions

A relation in mathematics is simply a set of ordered pairs. A relation becomes a function when each input relates to exactly one output.
This is a pivotal distinction because functions provide a specific relationship where no x-value repeats. This ensures every input leads to a unique output.
In our example, the relation \([(-3,-3),(-2,-2),(-1,-1),(0,0)]\) maps each domain value to one unique range value. Hence, it confirms as a function.
Remember, if an x-value appears more than once with different y-values, then it is not a function. However, in our scenario, no x-value repeats, maintaining the integrity of a function.

Ordered Pairs Analysis

Analyzing ordered pairs helps in decoding the nature of relations and functions.
Each ordered pair consists of two parts: (x, y) – the x-value (domain/input) and the y-value (range/output).
By examining pairs such as \((-3, -3), (-2, -2), (-1, -1), (0, 0)\), you can infer the mapping between x and y.
This one-to-one mapping indicates that each x corresponds to a single y. It's this analysis that allows us to conclude the relation as a function.
Breaking down ordered pairs into their components can simplify problem-solving in algebra, offering a clearer view of the function's behavior and ensuring deeper understanding.