Problem 3 - Algebra A Combined Function

Question

Find the domain and the range of each relation. Also determine whether the relation is a function. $$ \\{(-2,4),(6,4),(-2,-3),(-7,-8)\\} $$

Step-by-Step Solution

Answer

Domain: \{-2, 6, -7\}; Range: \{4, -3, -8\}; Not a function.

Step 1: Identify the domain

To find the domain of a relation, we look at all the first elements of the ordered pairs. In this relation, the ordered pairs are \((-2, 4), (6, 4), (-2, -3), (-7, -8)\). The domain is the set of all these first elements: \(-2, 6, -2, -7\). Since each element in the domain should be unique, the domain is \{-2, 6, -7\}.

Step 2: Identify the range

The range is the set of all second elements from the ordered pairs. Looking at the pairs \((-2, 4), (6, 4), (-2, -3), (-7, -8)\), the second elements are \(4, 4, -3, -8\). Again, ensuring uniqueness, the range is \{4, -3, -8\}.

Step 3: Determine if the relation is a function

A relation is a function if every element in the domain is paired with exactly one element in the range. Look at the domain values: \(-2\) pairs with \(4\) and \(-3\), and all other domain values are paired uniquely. Since \(-2\) corresponds to two different range values, the relation is not a function.

Key Concepts

Relation

A relation in mathematics is a connection between sets of information. It is a set of ordered pairs, where each pair consists of two elements. In the given exercise, the pairs are

• (-2, 4)
• (6, 4)
• (-2, -3)
• (-7, -8)

The first number in each pair is derived from the domain, and the second number is from the range.
Child's play, right? But every relation needs to be carefully looked at, especially when determining other properties such as the domain, range, and whether it is a function.

Function

A function is a special type of relation where each element in the domain pairs with exactly one element in the range. Basically, for every input, there can be only one output in a function.
Think of it like a funnel: it can handle lots of inputs, but for each distinct input, there can only be one output. In our example,

• The domain elements are: -2 occurs twice paired with 4 and -3
• All other elements uniquely relate.

Now, since -2 is paired with two outputs, it doesn't fit the mold of a function.

Ordered Pairs

Ordered pairs are the building blocks of relations and functions. Each ordered pair is an arrangement of two elements surrounded by parentheses, like (-2, 4).
The first element refers to a value in the domain, and the second element refers to a value in the range. It's essential for understanding relationships in mathematics.

• The order in ordered pairs is crucial
• Swapping the elements transforms the meaning entirely.

For example, if you swapped (-2, 4) to (4, -2), the context and relation would change too.