Problem 10 - Algebra A Combined Function

Question

Find the domain and the range of each relation. Also determine whether the relation is a function. $$ \\{(\pi, 0),(0, \pi),(-2,4),(4,-2)\\} $$

Step-by-Step Solution

Answer

Domain: \(\{\pi, 0, -2, 4\}\); Range: \(\{0, \pi, 4, -2\}\); Yes, it is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all possible input values (x-values). To find the domain, list all the first elements from each ordered pair in the relation. Here, the ordered pairs are \((\pi, 0), (0, \pi), (-2, 4), (4, -2)\). Thus, the domain is \(\{\pi, 0, -2, 4\}\).

Step 2: Identify the Range

The range of a relation is the set of all possible output values (y-values). To find the range, list all the second elements from each ordered pair in the relation. Here, the ordered pairs are \((\pi, 0), (0, \pi), (-2, 4), (4, -2)\). Therefore, the range is \(\{0, \pi, 4, -2\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if every input value (x-value) in the domain is associated with exactly one output value (y-value). Inspect each ordered pair in the given relation: \((\pi, 0), (0, \pi), (-2, 4), (4, -2)\). Each x-value is unique and does not repeat. Thus, the relation is a function.

Key Concepts

Domain and Range

In the realm of relations and functions, the concepts of domain and range play a pivotal role. The **domain** is simply the collection of all possible input values or x-values that a relation can have. When you are given a set of points, you extract the x-value from each ordered pair to make up the domain. For instance, with the ordered pairs \((\pi, 0), (0, \pi), (-2, 4), (4, -2)\), the domain consists of \(\{\pi, 0, -2, 4\}\). Each component is a potential input, and together they form the set of the domain.

The **range** mirrors the domain but applies to output values instead. This is the set of all y-values you can get from the relationship. Again, using our ordered pairs, you would select the y-value from each pair to build the range. Therefore, the range of the given relation is \(\{0, \pi, 4, -2\}\). This collection of values represents all the possible outcomes or y-values for the given inputs. Observing both domain and range allows us to understand the full scope of a relation's potential inputs and outputs.

Remember:

• Domain: Set of all x-values
• Range: Set of all y-values

Ordered Pairs

Ordered pairs are fundamental in describing relations and functions. They generally take the form \((x, y)\), where the first element, \(x\), is the input or the domain value, and the second element, \(y\), is the output or the range value. These pairs show a direct association between an x-value and a y-value, making the relationship between inputs and outputs clear.

In our example, we have the ordered pairs: \((\pi, 0), (0, \pi), (-2, 4), (4, -2)\). Each of these ordered pairs illustrates how an element from the domain is linked to an element in the range.

This concept is crucial because it ensures that the relationship between input and output is explicitly defined, which is essential when determining whether a relation is a function. If you map these pairs visually on a coordinate plane, they provide a clear and understandable way of seeing the relation's behavior.

Important Points:

• Ordered pairs consist of two components: \((x, y)\).
• The "order" refers to the position of x and y, which are crucial for identifying the values accurately.

Function Definition

A function is a special type of relation that adheres to stricter rules. For a relation to be classified as a function, every element in the domain must be paired with exactly one element in the range. It means that for each x-value, there is only one corresponding y-value. This ensures a well-defined and unique mapping from inputs to outputs.

In the given example, the ordered pairs \((\pi, 0), (0, \pi), (-2, 4), (4, -2)\) show that each x-value appears only once, with no repetitions. Hence, each domain element is paired with one and only one range element, satisfying the condition of a function.

Understanding whether a relation is a function can be visually checked with the "vertical line test" on a graph. If a vertical line crosses the graph more than once, then the relation is not a function.Key Takeaways:

• Each x-value in a function maps to one y-value.
• A non-repeating x-value confirms a relation is a function.
• Can use a vertical line test for a quick visual check.