Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-5 :\) a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the domain. c. List the clements of the range. $$ \\{(1,-1),(0,0),(1,1)\\} $$

Step-by-Step Solution

Answer

The set is not a function. Domain: \(\{0, 1\}\). Range: \(\{-1, 0, 1\}\).

Step 1: Check for uniqueness of x-values

A set of ordered pairs is a function if each x-value (input) is unique. Examine the set \(\{(1,-1),(0,0),(1,1)\}\). Here, the x-value 1 appears twice, with different y-values (-1 and 1). This violates the requirement for a function.

Step 2: List the domain

The domain of a relation is the set of all unique x-values. From the set \(\{(1,-1),(0,0),(1,1)\}\), the x-values are 1 and 0. Therefore, the domain is \(\{0, 1\}\).

Step 3: List the range

The range of a relation is the set of all y-values. In the set \(\{(1,-1),(0,0),(1,1)\}\), the y-values are -1, 0, and 1. Thus, the range is \(\{-1, 0, 1\}\).

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental concept in mathematics. They are used to describe relationships between two quantities, usually expressed as \((x, y)\). In a pair, \(x\) is the first element, and \(y\) is the second element. This order matters significantly, hence the name 'ordered pairs'. Each pair in a given set represents a point or a connection between two variables.
Ordered pairs are frequently used in functions, mappings, and graphing. It is essential to note that an ordered pair \((x, y)\) is different from \((y, x)\), because switching the values changes the meaning and relationship.
The concept of ordered pairs is crucial when determining whether a relation is a function or when identifying the domain and range. Identifying repetition or inconsistency in the x-values helps us determine functional relationships in mathematics.

Domain and Range Explained

When discussing functions, two crucial sets come into play: the domain and the range.
The domain of a function or relation consists of all possible input values, often represented by the variable \(x\). For our example, \(\{(1,-1),(0,0),(1,1)\}\), the domain is derived by examining all unique x-values. These are \(0\) and \(1\). If a value appears more than once as an x-value, it is only listed once in the domain.
The range, on the other hand, is the set of all possible output values or y-values. It represents all the results we get when we input the domain into our relation. From the same set of pairs, the range is \(\{-1, 0, 1\}\). Every y-value from the ordered pairs is included.
Understanding the domain helps in knowing the constraints and boundaries of input values, while the range provides insight into potential outputs or results.

What Makes a Function?

A function in mathematics is a specific type of relation. To qualify as a function, each input value should have exactly one output. This means that in a set of ordered pairs, the x-values must be unique.

• If an x-value repeats with a different y-value, the set cannot be a function.
• Functions are a core concept because they allow one-to-one relationships whereas a mere relation might correspond to multiple outputs.

In the provided example set \(\{(1,-1),(0,0),(1,1)\}\), the x-value \(1\) is associated with two different y-values: \(-1\) and \(1\). This breaks the function criteria, which requires every x-value to pair with just one y-value.
This criterion ensures a consistent and predictable relationship, fundamental to many areas of math, including calculus and algebra. Recognizing when a relation is a function is an initial step that helps in understanding more complex mathematical structures.