Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(5,1),(3,2),(4,9),(7,6)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{5, 3, 4, 7\}\); Range: \(\{1, 2, 9, 6\}\); The relation is a function.

Step 1: Identifying the Domain

To identify the domain of the relation, we must take all the first elements from each ordered pair. These are the x-values or inputs. For the set \((5, 1), (3, 2), (4, 9), (7, 6)\), the domain is \(D = \{5, 3, 4, 7\}\).

Step 2: Identifying the Range

The range consists of the second elements from each ordered pair, which are the y-values or outputs. For the set \((5, 1), (3, 2), (4, 9), (7, 6)\), the range is \(R = \{1, 2, 9, 6\}\).

Step 3: Checking if the Relation is a Function

A relation is a function if each input in the domain maps to exactly one output in the range. Examine each x-value: 5, 3, 4, and 7 each map to only one unique y-value (1, 2, 9, and 6 respectively). Since no x-value is associated with more than one y-value, this relation is a function.

Key Concepts

Understanding Domain in Functions

The domain of a function is a crucial concept in mathematics.
It represents all the possible input values that a function can accept. In simpler terms, think of the domain as the set of all x-values in your ordered pairs.
In our original exercise, the ordered pairs are \((5, 1), (3, 2), (4, 9), (7, 6)\). To find the domain, you identify the first number in each set of ordered pair:

• 5 from (5, 1)
• 3 from (3, 2)
• 4 from (4, 9)
• 7 from (7, 6)

Hence, the domain \(D\) is \(\{5, 3, 4, 7\}\).
These values indicate all possible inputs for the function. Having the domain means you know the limits on what x-values can be used.
These values direct where you focus when solving or graphing functions.

Exploring the Range of Functions

The range of a function is equally important.
It involves all possible outputs the function can produce based on the domain.
Within ordered pairs, the range corresponds to the second number in each pair—the y-values. From the exercise's set \((5, 1), (3, 2), (4, 9), (7, 6)\):

• 1 from (5, 1)
• 2 from (3, 2)
• 9 from (4, 9)
• 6 from (7, 6)

This gives us the range \(R = \{1, 2, 9, 6\}\).
The range outlines the limits of a function's output.
Understanding it helps predict the function's behavior and results.

Demystifying Ordered Pairs

Ordered pairs play a foundational role in understanding functions.
Each ordered pair is expressed as \((x, y)\), where 'x' is the input, and 'y' is the output. In our exercise, we have the ordered pairs:

• (5, 1)
• (3, 2)
• (4, 9)
• (7, 6)

Ordered pairs allow us to visually represent relations between different elements.
By plotting these pairs, you can easily identify patterns or trends. Furthermore, recognizing that a relation is a function depends on how these pairs are related.
A relation is a function if every x-value is paired with exactly one y-value.
In our example, each x:

• 5,
• 3,
• 4,
• 7,

corresponds to a unique y:

• 1,
• 2,
• 9,
• 6 respectively.

This confirms that our relation is indeed a function. Understanding ordered pairs ensures you know how inputs relate to outputs within a function or relation.