Problem 1 - College Algebra Essentials

Question

Determine whether each relation is a function. Give the domain and range for each relation. $$ \\{(1,2),(3,4),(5,5)\\} $$

Step-by-Step Solution

Answer

The given relation \{(1,2),(3,4),(5,5)\} is indeed a function. The domain of this function is \{1, 3, 5\}, and the range is \{2, 4, 5\}.

Step 1: Determine whether the relation is a function

Analyze the order pairs to determine whether the relation is a function by inspecting that every x-value is paired with exactly one y-value. In this case, each x-value in set \{(1,2),(3,4),(5,5)\} corresponds to only one y-value. So, the relation is a function.

Step 2: Identify the Domain

The domain of a function is the set of all possible x-values. Look at the given pairs and identify the x-values. In this case, the domain is \{1, 3, 5\}.

Step 3: Identify the Range

The range of a function is the set of all possible y-values, which are output of the function. In our case, the range is \{2, 4, 5\}.

Key Concepts

Understanding Domain and Range

The concept of domain and range is fundamental when dealing with functions in algebra. Let's break these down:

The **domain** of a function consists of all the possible input values, often represented as 'x'. For any function or relation, the domain is simply the set of all x-values that you can substitute into the function. In our given relation \( \{ (1,2), (3,4), (5,5) \} \), the domain is derived from listing all the x-values from each ordered pair, which gives us \( \{ 1, 3, 5 \} \).

Moving on to the **range**, this is the set of all potential output values 'y' produced by the function. By looking at our set of ordered pairs, we derive the range by identifying all the y-values, giving us \( \{ 2, 4, 5 \} \).

It is essential to remember:

• The domain is linked to possible input values or "what goes into the function."
• The range comprises all possible output values or "what you get out of the function."

Identifying domain and range helps determine how broad or limited a function's set of solutions can be.

Relations in Mathematics

In mathematics, a relation is a set of ordered pairs, which essentially links elements from two sets. Relations serve as the foundation for understanding functions, as every function is a type of relation.

A **relation** can connect any two elements from different sets or even the same set, in various configurations. Each pair in our relation example, \( \{ (1,2), (3,4), (5,5) \} \), links an element from the domain (x-values) to an element in the range (y-values).

Relations can be depicted in several ways such as:

• **Graphically** - Often using a graph of points on the coordinate plane.
• **Algebraically** - Using equations or set notations like our example.

To determine if a relation is a **function**, it must pair each input with only one output. In our instance, each x-value from \( \{ 1,3,5 \} \) only appears once. That means this specific relation qualifies as a function, illustrating this central concept in mathematics.

Exploring Ordered Pairs

In algebra, an **ordered pair** is a pair of numbers that are written in a specific order, typically as \( (x, y) \). This pairing is crucial for defining relationships between numbers, often in the context of coordinate systems.

Ordered pairs are used to plot points on a graph and represent solutions for equations, showing how input values relate to output values. In our set of ordered pairs \( \{ (1,2), (3,4), (5,5) \} \):

• The first element 'x' denotes the horizontal position or input value.
• The second element 'y' reflects the vertical position or output value.

Understanding how to work with ordered pairs is important for analyzing graphs and equations. Notably, the sequence in ordered pairs is significant because reversing them would represent different relationships and positions on a graph.

Ordered pairs form the building blocks of coordinate geometry, allowing for the visual representation of functions and relations, thus aiding in their analysis and understanding.