Problem 8 - Algebra A Combined Function

Question

Find the domain and the range of each relation. Also determine whether the relation is a function. $$ \left\\{\left(\frac{1}{2}, \frac{1}{4}\right),\left(0, \frac{7}{8}\right),(0.5, \pi)\right\\} $$

Step-by-Step Solution

Answer

Domain: \(\{\frac{1}{2}, 0, 0.5\}\). Range: \(\{\frac{1}{4}, \frac{7}{8}, \pi\}\). Yes, it's a function.

Step 1: Identify the Domain

The domain of a relation consists of all the first elements of the ordered pairs. In this case, the ordered pairs are \(\left( \frac{1}{2}, \frac{1}{4} \right)\), \(\left(0, \frac{7}{8} \right)\), and \((0.5, \pi)\). Thus, the domain is \( \left\{ \frac{1}{2}, 0, 0.5 \right\} \).

Step 2: Identify the Range

The range of a relation consists of all the second elements from the ordered pairs. Here, the ordered pairs are \(\left( \frac{1}{2}, \frac{1}{4} \right)\), \(\left(0, \frac{7}{8} \right)\), and \((0.5, \pi)\). So, the range is \( \left\{ \frac{1}{4}, \frac{7}{8}, \pi \right\} \).

Step 3: Determine if the Relation is a Function

A relation is a function if each element in the domain is mapped to exactly one element in the range. Here, each domain element \( \frac{1}{2}, 0, \) and \( 0.5 \) maps to only one value in the range. Therefore, this relation is a function.

Key Concepts

Relation

In mathematics, a **relation** is simply a collection of ordered pairs. Each pair consists of two elements and represents a unique association between them.

Consider the provided set of ordered pairs from the exercise:

• \( \left( \frac{1}{2}, \frac{1}{4} \right) \)
• \( \left( 0, \frac{7}{8} \right) \)
• \( (0.5, \pi) \)

This is a relation because it groups several pairs of numbers together. The domain refers to the first element in each pair, and the range refers to the second element. Understanding relations helps in identifying how different elements or sets relate to each other. Relations can be analyzed further to determine if they have specific characteristics like being functions or if they have symmetric, reflexive, or transitive properties.

For any given relation, knowing how elements connect allows us to explore further mathematical concepts, such as domains and ranges.

Function

A **function** is a special kind of relation. What makes a relation a function is that each element in its domain is paired with exactly one element in its range.

In simpler terms, a function has an input (the domain) and an output (the range), with a one-to-one mapping from each input to a specific output. Let's look at the relation in our exercise:

• Each element of the domain \( \left\{ \frac{1}{2}, 0, 0.5 \right\} \) maps uniquely to elements in the range \( \left\{ \frac{1}{4}, \frac{7}{8}, \pi \right\} \).

This is why the given relation is a function. There are no repeated domain elements being assigned to different range elements.

Functions are crucial in mathematics because they represent consistent rules or laws. Every time you have a function, you know that the function will act `reliably` on each input.

Ordered Pairs

An **ordered pair** consists of two elements where the order in which the elements are placed is important. It is written in the form \((x, y)\), where \(x\) and \(y\) can represent any number or object. In our exercise, the ordered pairs are:

• \( \left( \frac{1}{2}, \frac{1}{4} \right) \)
• \( \left(0, \frac{7}{8} \right) \)
• \( (0.5, \pi) \)

Ordered pairs are fundamental in defining relations and functions because they lay out the precise connection between the input and output with clarity. The first number in the pair (\(x\)) is typically considered the input or the domain element, while the second number (\(y\)) is the output or range element.

Ordered pairs are often plotted on a two-dimensional graph where \(x\) values are on the horizontal axis and \(y\) values are on the vertical axis. Recognizing the importance of ordered pairs helps in visualizing and analyzing mathematical relationships.