Problem 8 - Precalculus

Question

Determine whether or not the relation represents \(y\) as a function of \(x .\) Find the domain and range of those relations which are functions. $$ \\{\ldots,(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9), \ldots\\} $$

Step-by-Step Solution

Answer

The relation represents \(y\) as a function of \(x\). Domain: all integers. Range: non-negative perfect squares.

Step 1: Check for Function

To determine if a relation is a function, each input must map to exactly one output. Analyze the pairs given: - From the set \{...(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9),...\}, each distinct \(x\) value has one corresponding \(y\) value. - Thus, this relation is a function because each \(x\)-value has a unique \(y\)-value.

Step 2: Determine the Domain

The domain of a function is the set of all possible input values \(x\). Since the relation includes pairs like \((-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9)\) and this is implicit for all integers (as evident from the pattern and ellipsis), the domain is all integers or \( ext{Domain} = \{ x \in \mathbb{Z} \} \).

Step 3: Determine the Range

The range of a function is the set of all possible output values \(y\). From the pairs \{...(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9),...\}, the outputs are perfect squares: 0, 1, 4, 9, and so on (corresponding to squares of integers). Thus, the range is non-negative integers or \( ext{Range} = \{ y \in \mathbb{Z} | y = n^2, n \in \mathbb{Z} \} \).

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of input values (the "x" values) for which the function is defined. When determining the domain, think about all possible ways the input variable can exist. For our specific relation, represented as

• \{...(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9),...\}

you will notice that every integer can act as an input. This pattern indicates that the domain includes all integers. Thus, if you imagine an endless line of integer values marching along the x-axis, this line represents all the possible inputs to the function. Therefore, we express the domain as:

• \(\text{Domain} = \{ x \in \mathbb{Z} \}\)

Range of a Function

The range of a function is the set of all possible output values (the "y" values) that a function can produce from its domain. Analyzing the outputs in the given relation yields perfect squares: 0, 1, 4, 9, and so forth.These outputs, produced by squaring each integer input, suggest a specific pattern. The range only includes non-negative integers that are perfect squares. This is consistent because squaring an integer or any number always results in a non-negative output. Thus, the range of this function can be described as:

• \(\text{Range} = \{ y \in \mathbb{Z} \mid y = n^2, \; n \in \mathbb{Z} \} \)

The notation \(y = n^2\) emphasizes that the range comprises all perfect squares.

Integer Functions

Integer functions involve operations where inputs and outputs are restricted to integer values. In the context of this problem, we see an integer function depicted through the relation

• \{...(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9),...\}.

Such functions are fascinating as they neatly capture patterns and relationships found in the realm of whole numbers, without decimals or fractions.In our example, each integer input outputs an integer value that is a perfect square, a specific type of integer function that highlights the beauty of these numbers:

• The input \(x\) values are integers.
• The output \(y\) values, \(y = x^2\), are also integers.

Recognizing these functions can help simplify problems. It highlights how powerful the relationships in integer functions can be, as they fundamentally rely on whole number interactions.