Problem 6 - 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. $$ \\{(x, 1) \mid x \text { is an irrational number }\\} $$

Step-by-Step Solution

Answer

Relation is a function; domain: irrational numbers; range: \(\{1\}\).

Step 1: Understanding the Relation

The relation is given as \( \{(x, 1) \mid x \text{ is an irrational number} \} \). This means for each irrational number \(x\), the corresponding \(y\)-value is constantly \(1\).

Step 2: Evaluating Function Criteria

To determine if a relation represents \(y\) as a function of \(x\), each \(x\) must correspond to exactly one \(y\). Here, each irrational \(x\) has a unique and consistent \(y\) value of 1.

Step 3: Conclusion on Function Nature

Since for every \(x\) there is a unique \(y\) value (which is 1), the relation is a function.

Step 4: Determining the Domain

The domain of the relation, since it includes all irrational numbers, is the set of all irrational numbers \(x\).

Step 5: Determining the Range

The range is determined by the possible \(y\)-values. In this relation, the \(y\)-value is consistently 1, thus the range is \(\{1\}\).

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial in mathematics. The domain is essentially all the possible input values (often represented by "x") that a function can accept without leading to undefined operations. For the relation given, our domain consists of all irrational numbers.
Irrational numbers are real numbers that cannot be expressed as a simple fraction, like \( equire{cancel} \ \sqrt{2} \) or \(\pi\). In this context, each irrational number \(x\) maps to a single \(y\)-value of 1. Thus, the domain includes all irrational numbers.

• Domain: all irrational numbers
• Range: the set of all possible outputs ("y" values), so here it's \(\{1\}\)

The range refers to all the possible values that come out of the function after substituting the domain values. Here, no matter which irrational number you pick as \(x\), you always get 1 as the result. Therefore, the range is just the number \(1\).

Irrational Numbers

Irrational numbers are numbers that cannot be precisely written as a ratio of two integers. They have non-repeating and non-terminating decimal expansions. Famous examples include \(\sqrt{2}\), \(\pi\), and \(e\). These numbers cannot be expressed as a simple fraction, which makes them irrational.

• Non-repeating: Their decimal representation never takes on a repeating pattern.
• Non-terminating: They continue indefinitely without any end.

The concept of irrational numbers is fundamental in understanding various mathematical and real-world phenomena. In functions and relations, irrational numbers often show up in domains, as they do in the exercise you're studying. Appreciating their unique properties helps in grasping how they behave in different scenarios.

Relations and Functions

Not every relation is a function, but every function is a relation. Let's unpack why that is. In a mathematical relation, we are exploring a connection between two sets of values, typically called the domain and range. When every element in the domain pairs with exactly one element in the range, we have a function.

• Function: Each input has a unique output.
• Relation: A set of ordered pairs that associates inputs with outputs.

The exercise illustrates a scenario where each irrational number \(x\) always maps to the \(y\)-value of 1. This meets the "unique output" condition of a function. Even if you choose different irrational numbers as input, you consistently receive \(1\) as the output, validating the relation as a function. Understanding the definition and difference between a relation and a function helps in navigating more complex topics in mathematics.