range of a relation

Range of a Relation in Math

Definition

The range of a relation in math refers to the set of all possible output values or the set of all values that the dependent variable can take on. In simpler terms, it is the collection of all the y-values that correspond to the x-values in a relation.

History

The concept of range has been a fundamental part of mathematics for centuries. It can be traced back to ancient civilizations such as the Babylonians and Egyptians, who used mathematical principles to solve practical problems. However, the formal definition and study of range as a mathematical concept emerged during the development of set theory in the late 19th and early 20th centuries.

Grade Level

The concept of range of a relation is typically introduced in middle school or early high school mathematics. It is an important topic in algebra and functions, as it helps students understand the behavior and limitations of a given relation.

Knowledge Points and Explanation

To understand the range of a relation, one must first grasp the concept of a relation itself. A relation is a set of ordered pairs, where each ordered pair consists of an input value (x) and an output value (y). The range of a relation is the set of all possible y-values that correspond to the x-values in the relation.

To find the range of a relation, follow these steps:

1. Identify all the y-values in the relation.
2. Remove any duplicate values.
3. Arrange the remaining values in ascending order.

Types of Range of a Relation

There are two main types of range of a relation:

1. Finite Range: When the set of possible output values is limited and can be listed explicitly, the range is said to be finite.
2. Infinite Range: When the set of possible output values is unbounded and cannot be listed explicitly, the range is said to be infinite.

Properties of Range of a Relation

The range of a relation possesses several properties:

1. It can be a single value, a set of values, or even an empty set.
2. It is always a subset of the codomain, which is the set of all possible output values.
3. It can be equal to or smaller than the codomain, but it cannot be larger.

Finding the Range of a Relation

To find or calculate the range of a relation, follow these steps:

1. Identify all the y-values in the relation.
2. Remove any duplicate values.
3. Arrange the remaining values in ascending order.

Formula or Equation for Range of a Relation

There is no specific formula or equation to calculate the range of a relation. It depends on the given relation and the set of possible output values.

Applying the Range of a Relation Formula or Equation

As mentioned earlier, there is no specific formula or equation for the range of a relation. Instead, it requires identifying and organizing the y-values in the relation.

Symbol or Abbreviation for Range of a Relation

The symbol used to represent the range of a relation is "R".

Methods for Range of a Relation

There are several methods to determine the range of a relation, including:

1. Listing all the y-values and removing duplicates.
2. Graphing the relation and identifying the y-values.
3. Using algebraic techniques to solve for the output values.

Solved Examples on Range of a Relation

1. Given the relation {(1, 2), (3, 4), (5, 6)}, find the range. Solution: The range is {2, 4, 6}.

2. Find the range of the relation y = x^2. Solution: The range is all non-negative real numbers, [0, ∞).

3. Determine the range of the relation y = 3x - 2. Solution: The range is all real numbers, (-∞, ∞).

Practice Problems on Range of a Relation

1. Find the range of the relation {(2, 5), (4, 7), (6, 5), (8, 9)}.
2. Determine the range of the relation y = 2x + 3.
3. Given the relation {(1, 3), (2, 5), (3, 7)}, find the range.

FAQ on Range of a Relation

Question: What is the range of a relation? Answer: The range of a relation refers to the set of all possible output values or the set of all values that the dependent variable can take on.