Domain Of Cube Root Function W(x) = ∛(7x + 23) In Interval Notation

## Introduction: Understanding the Domain

In the realm of mathematics, the **domain** of a function is a fundamental concept. It defines the set of all possible input values (often represented by 'x') for which the function produces a valid output. Determining the domain is crucial for understanding the behavior and limitations of a function. This article delves into the process of finding the domain of a specific function: W(x) = ∛(7x + 23). This function involves a cube root, which has unique properties that affect its domain. We will explore these properties and apply them to determine the domain of W(x), expressing our answer in interval notation. By the end of this guide, you'll have a solid understanding of how to find the domain of cube root functions and similar mathematical expressions.

## Demystifying Cube Root Functions

To effectively determine the domain of W(x) = ∛(7x + 23), we must first understand the nature of **cube root functions**. Unlike square roots, which only accept non-negative inputs to produce real number outputs, cube roots can handle both positive and negative inputs. This is because a negative number multiplied by itself three times results in a negative number. For example, ∛(-8) = -2, since (-2) * (-2) * (-2) = -8. This key characteristic of cube roots significantly impacts the domain of functions that contain them.

Because cube roots can accept any real number as input, the expression inside the cube root, in this case, 7x + 23, can be any real number. This is a crucial distinction from functions involving square roots or other even-indexed roots, where the radicand (the expression inside the root) must be greater than or equal to zero. Understanding this difference is the first step in correctly identifying the domain of our function W(x). In the following sections, we will leverage this understanding to formally determine the domain of W(x) and express it in interval notation.

## Determining the Domain of W(x) = ∛(7x + 23)

Now, let's apply our understanding of cube root functions to the specific function W(x) = ∛(7x + 23). As we established, the expression inside the cube root, 7x + 23, can be any real number. This means there are no restrictions on the values that x can take. To find the domain, we need to identify any values of x that would make the function undefined. For cube root functions, there are no such values. The cube root of any real number, whether positive, negative, or zero, is a real number. This is a fundamental property of cube roots that sets them apart from square roots and other even-indexed roots.

Since 7x + 23 can be any real number, x itself can also be any real number. There are no values of x that would cause the function to be undefined. Therefore, the domain of W(x) is the set of all real numbers. This is a critical conclusion that stems directly from the properties of cube root functions. In the next section, we will express this domain using interval notation, a standard way to represent sets of numbers.

## Expressing the Domain in Interval Notation

Having determined that the domain of W(x) = ∛(7x + 23) is all real numbers, we now need to express this domain in **interval notation**. Interval notation is a concise way to represent a set of numbers using intervals. It uses parentheses and brackets to indicate whether the endpoints of an interval are included or excluded. Parentheses indicate exclusion, while brackets indicate inclusion.

To represent all real numbers in interval notation, we use the symbols -∞ (negative infinity) and ∞ (positive infinity). These symbols represent the unbounded nature of the real number line, extending infinitely in both the negative and positive directions. Since infinity is not a number but rather a concept of unboundedness, we always use parentheses with infinity symbols.

Therefore, the domain of W(x) = ∛(7x + 23), which includes all real numbers, is expressed in interval notation as (-∞, ∞). This notation signifies that the function is defined for any value of x between negative infinity and positive infinity, encompassing all real numbers. This is a direct consequence of the properties of cube root functions, which can accept any real number as input.

## Conclusion: The Unrestricted Domain of Cube Root Functions

In conclusion, we have successfully determined the domain of the function W(x) = ∛(7x + 23). By understanding the fundamental properties of **cube root functions**, we recognized that the expression inside the cube root, 7x + 23, can be any real number. This led us to the crucial realization that there are no restrictions on the values of x, meaning the domain encompasses all real numbers. We then expressed this domain concisely and accurately using interval notation as (-∞, ∞).

This exploration highlights the importance of understanding the characteristics of different types of functions when determining their domains. While functions involving square roots or other even-indexed roots have restrictions due to the need for non-negative radicands, cube root functions offer a broader scope, accepting any real number as input. This knowledge is essential for accurately analyzing and interpreting mathematical functions. By mastering these concepts, you can confidently tackle domain-related problems and gain a deeper understanding of mathematical functions.

### Key Takeaways:

*   The **domain** of a function is the set of all possible input values (x) for which the function produces a valid output.
*   **Cube root functions** can accept any real number as input, unlike square roots which only accept non-negative inputs.
*   For W(x) = ∛(7x + 23), the expression inside the cube root (7x + 23) can be any real number.
*   Therefore, the domain of W(x) is all real numbers.
*   In **interval notation**, the domain of W(x) is expressed as (-∞, ∞).

This comprehensive guide has provided a clear and detailed explanation of how to determine the domain of a cube root function. By understanding the principles outlined in this article, you can confidently approach similar problems and enhance your mathematical skills.