Domain Of Cube Root Function W(x) = ³√(7x + 23) In Interval Notation

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Introduction

In the fascinating world of mathematics, functions serve as fundamental building blocks, mapping inputs to outputs and describing relationships between variables. Understanding the domain of a function is crucial, as it defines the set of all possible input values for which the function produces a valid output. In this article, we will embark on a comprehensive exploration of the function W(x) = ³√(7x + 23), delving into the intricacies of determining its domain and expressing it in interval notation. We will unravel the concepts of cube roots, real numbers, and the constraints that dictate the domain of radical functions. So, let's dive into the captivating realm of cube root functions and discover the domain of W(x) = ³√(7x + 23).

Understanding the Domain of a Function

Before we embark on our journey to determine the domain of W(x) = ³√(7x + 23), it's essential to establish a firm understanding of what the domain of a function truly represents. In essence, the domain of a function is the set of all possible input values, commonly denoted as 'x', for which the function produces a valid and real-valued output. Imagine a function as a machine that transforms inputs into outputs. The domain comprises all the inputs that the machine can process without encountering any errors or undefined results.

For instance, consider a simple function like f(x) = 1/x. In this case, the domain consists of all real numbers except for x = 0, because division by zero is undefined in mathematics. Similarly, for functions involving square roots, the domain is restricted to non-negative values, as the square root of a negative number is not a real number. When determining the domain of a function, we must carefully consider any restrictions imposed by the function's structure, such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

Delving into Cube Root Functions

Now that we have a solid grasp of the concept of a function's domain, let's turn our attention to the specific type of function we're interested in: cube root functions. A cube root function is a mathematical expression that involves taking the cube root of a variable or an expression. The cube root of a number is the value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8.

Cube root functions possess a unique characteristic that sets them apart from square root functions and other even-indexed root functions. Unlike square roots, which are only defined for non-negative numbers, cube roots can accept both positive and negative numbers as inputs. This stems from the fact that a negative number multiplied by itself three times results in a negative number. For instance, the cube root of -8 is -2, because (-2) × (-2) × (-2) = -8. This crucial property of cube roots significantly influences the domain of functions involving them.

Analyzing W(x) = ³√(7x + 23)

With a clear understanding of cube root functions, we can now focus on the function at hand: W(x) = ³√(7x + 23). This function takes an input 'x', multiplies it by 7, adds 23, and then calculates the cube root of the resulting expression. Our mission is to determine the set of all possible 'x' values for which this function produces a valid real number output. To accomplish this, we must carefully consider any potential restrictions imposed by the cube root operation.

As we established earlier, cube roots are defined for all real numbers, both positive and negative. This implies that the expression inside the cube root, which is (7x + 23) in this case, can take on any real value without causing the function to become undefined. Therefore, there are no inherent restrictions on the values that 'x' can assume. This crucial observation forms the foundation for determining the domain of W(x) = ³√(7x + 23).

Determining the Domain of W(x)

Based on our analysis, we've concluded that the expression inside the cube root, (7x + 23), can be any real number. This means that there are no restrictions on the values that 'x' can take. In other words, 'x' can be any real number, and the function W(x) will still produce a valid real number output. To express this mathematically, we say that the domain of W(x) is the set of all real numbers.

Expressing the Domain in Interval Notation

Now that we've determined the domain of W(x) = ³√(7x + 23), we can express it using interval notation, a concise and widely accepted way of representing sets of numbers. Interval notation employs parentheses and brackets to indicate whether the endpoints of an interval are included or excluded. Parentheses signify that the endpoint is not included, while brackets indicate that it is included. The symbols -∞ and ∞ represent negative infinity and positive infinity, respectively, and are always enclosed in parentheses, as they are not actual numbers but rather concepts.

In our case, the domain of W(x) encompasses all real numbers, extending from negative infinity to positive infinity. Therefore, in interval notation, we express the domain as (-∞, ∞). This notation signifies that 'x' can be any number between negative infinity and positive infinity, including all the numbers in between. This representation succinctly captures the fact that there are no restrictions on the input values for the function W(x) = ³√(7x + 23).

Conclusion

In this comprehensive exploration, we have successfully determined the domain of the function W(x) = ³√(7x + 23). By delving into the characteristics of cube root functions and analyzing the expression within the cube root, we concluded that the domain of W(x) encompasses all real numbers. We then expressed this domain in interval notation as (-∞, ∞), providing a clear and concise representation of the function's permissible input values.

Understanding the domain of a function is paramount in mathematics, as it dictates the boundaries within which the function operates meaningfully. By mastering the concepts and techniques involved in domain determination, we equip ourselves with a valuable tool for analyzing and interpreting mathematical functions. As you continue your mathematical journey, remember that the domain is the foundation upon which the rest of the function's behavior is built. By carefully considering the domain, you can gain a deeper understanding of the function's properties and its role in the broader mathematical landscape.