Problem 10 - 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. $$ \\{(x, 3) \mid-2 \leq x<4\\} $$

Step-by-Step Solution

Answer

Yes, it is a function. Domain: \([-2, 4)\); Range: \(\{3\}\).

Step 1: Define the Relation

We are given the relation \(\{(x, 3) \mid -2 \leq x < 4\}\). This means every input \(x\) within the interval \([-2, 4)\) has an output value of 3.

Step 2: Check if y is a Function of x

For \(y\) to be a function of \(x\), each \(x\) value must correspond to exactly one \(y\) value. Here, each \(x\) is paired with the same \(y\) value, 3. Therefore, \(y\) is a function of \(x\).

Step 3: Determine the Domain

The domain is the set of all possible \(x\) values that satisfy the relation. According to the problem, \(x\) ranges from \(-2\) to just less than \(4\), thus the domain is \([-2, 4)\).

Step 4: Determine the Range

The range is the set of all possible \(y\) values that result from the values in the domain. In this relation, every \(y\) is 3, so the range is \(\{3\}\).

Key Concepts

Domain and Range

Understanding the domain and range of a function is essential for grasping how functions work. The **domain** of a function consists of all the possible input values (typically represented by the variable \(x\)) that the function can accept. In our example, the domain is the set of all \(x\) values that satisfy the condition \(-2 \leq x < 4\). This means that \(x\) can be any number from \(-2\) up to, but not including, 4. It's expressed as the interval \([-2, 4)\).

The **range**, on the other hand, is the set of all possible output values (usually represented by \(y\)) that the function can produce. For the relation given, every \(x\) corresponds to the same output value, 3. Thus, the range is just the single value \(\{3\}\).

In summary, the domain defines where the function exists and the range specifies the values it can produce.

Relation

Let's explore what a relation is. In mathematics, a **relation** is simply a set of ordered pairs. Our example, \(\{(x, 3) \mid -2 \leq x < 4\}\), illustrates a relation where every \(x\) is paired with a \(y\) value of 3. This collection of pairs connects each element of the domain with an element in the range.

An important thing to note is that not all relations are functions. A **function** is a special type of relation where each input is related to exactly one output. In this case, each \(x\) corresponds to one \(y\), meaning this relation is indeed a function. This characteristic is key when discerning functions from mere relations.

Input and Output Values

Often in functions, we talk about **input and output values**. The input is what you give the function, and the output is what you get back. In terms of our relation \(\{(x, 3) \mid -2 \leq x < 4\}\), \(x\) is the input value, while \(y\), which is always 3, is the output value.

For this specific relation, every input value between -2 and less than 4 results in an output of 3. This regularity and predictability define what makes a function so reliable and useful in various applications. By understanding the inputs and their corresponding outputs, we comprehensively understand how the function behaves across its domain. So, whenever you see a relation, consider how the inputs and outputs are paired, and remember that concise pairing is what makes functions special!