Problem 1 - Algebra A Combined Function

Question

Find the domain and the range of each relation. Also determine whether the relation is a function. $$ \\{(-1,7),(0,6),(-2,2),(5,6)\\} $$

Step-by-Step Solution

Answer

Domain: \{-1, 0, -2, 5\}; Range: \{7, 6, 2\}; The relation is a function.

Step 1: Identify the Domain

The domain of a relation is the set of all first elements (or x-values) from each ordered pair. Here, we extract the x-values: \(-1, 0, -2, 5\). So, the domain is \(\{-1, 0, -2, 5\}\).

Step 2: Identify the Range

The range of a relation is the set of all second elements (or y-values) from each ordered pair. From the given relation, the y-values are \(7, 6, 2\). Since the value \(6\) appears twice but is only listed once in a set, the range is \(\{7, 6, 2\}\).

Step 3: Determine if the Relation is a Function

A relation is a function if each x-value in the domain is associated with exactly one y-value in the range. From the ordered pairs \((-1,7), (0,6), (-2,2), (5,6)\), all x-values \(-1, 0, -2, 5\) pair with unique y-values. No x-value is repeated, so the relation is a function.

Key Concepts

Relations

In mathematics, a relation is a collection of ordered pairs, essentially a set of inputs paired with outputs. Each pair consists of an x-value and a y-value. Understanding relations is crucial because they form the foundation for more complex concepts like functions.
For example, consider the relation \( \{(-1,7),(0,6),(-2,2),(5,6)\} \). This is simply a set of four ordered pairs. Each pair represents a point on a graph, potentially describing how certain quantities relate to each other.

• Ordered Pairs: These are formed by taking one value from the domain and one from the range.
• Set Notation: Relations are often presented in a curly-brace format which signifies a set of elements.

Whether these relations form a function or not depends on the uniqueness of the y-values for each x-value. This brings us to the discussion on functions.

Functions

A function is a special type of relation that pairs each x-value with exactly one y-value. This means for every input, there is a single, unambiguous output. Functions are incredibly important in mathematics because they form consistent and predictable patterns.
To determine if a relation, like our example relation \( \{(-1,7),(0,6),(-2,2),(5,6)\} \), is a function, we check that each x-value maps to one, and only one, y-value. In our example, since each x-value is unique and pairs with a unique y-value, this relation is indeed a function.

• Every x-value must map to exactly one y-value.
• No x-value is repeated with a different y-value.

Functions can be identified easily by listing all the x-values and checking for duplicates.

x-values

The x-values in a relation are the first numbers in each ordered pair. They are also referred to as the domain of the relation. These values are critical because they represent the inputs for which the outputs (y-values) are determined.
For instance, in \( \{(-1,7),(0,6),(-2,2),(5,6)\} \), the x-values are \(-1, 0, -2, 5\). These values form the set \(\{-1, 0, -2, 5\}\), which means any operations or functions applied will consider these as starting points.

• The domain is the set of all x-values for a relation.
• It is crucial that each x-value is unique in a function.

The uniqueness of x-values ensures that the function's rules are consistently applied, leading to predictable outcomes.

y-values

In a relation, the y-values are the second numbers in each ordered pair, and they make up what is known as the range. These values signify the outputs or results of applying the function's rule to each x-value.
Consider our example relation \( \{(-1,7),(0,6),(-2,2),(5,6)\} \). Here, the y-values are \(7, 6, 2\). Even though \(6\) appears twice, it is only counted once for the range \(\{7, 6, 2\}\).

• The range is the set of all y-values a relation can produce.
• Repeated y-values do not affect the determination of a function.

Understanding y-values and their behavior helps us comprehend what outcomes are possible from the given inputs.