Finding The Domain Of Cube Root Function F(x) = \sqrt[3]{5x + 25}

Jul 12, 2025 by ADMIN 66 views

$Find the Domain of the Cube Root Function f(x) = \sqrt[3]{5x + 25}$

Finding the domain of a function is a fundamental concept in mathematics, particularly in calculus and real analysis. The domain of a function represents the set of all possible input values (x-values) for which the function produces a valid output. Understanding the domain is crucial for analyzing the behavior of the function and its graphical representation. In this comprehensive guide, we will explore how to determine the domain of the function ${ f(x) = \sqrt[3]{5x + 25} }$ . We will delve into the properties of cube root functions and their implications for domain determination. By the end of this discussion, you will have a clear understanding of why the domain of this particular function is all real numbers.

Understanding Domain and Cube Root Functions

Before we dive into the specifics of the function ${ f(x) = \sqrt[3]{5x + 25} }$ , let's establish a solid understanding of the key concepts involved:

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real number output. In simpler terms, it's the collection of x-values that you can plug into the function without causing any mathematical errors. Common restrictions on the domain arise from:

Division by zero: A function is undefined when the denominator is zero.
Square roots of negative numbers: The square root of a negative number is not a real number.
Logarithms of non-positive numbers: Logarithms are only defined for positive arguments.

Cube Root Functions

A cube root function is a function of the form ${ f(x) = \sqrt[3]{g(x)} }$ , where ${ g(x) }$ is an expression involving ${ x }$ . The key characteristic of cube root functions is that they are defined for all real numbers. Unlike square roots, which only accept non-negative inputs, cube roots can handle both positive and negative inputs.

To illustrate, consider the basic cube root function ${ f(x) = \sqrt[3]{x} }$ . We can plug in any real number for ${ x }$ , and we will get a real number output. For instance:

${ \sqrt[3]{8} = 2 }$
${ \sqrt[3]{-8} = -2 }$
${ \sqrt[3]{0} = 0 }$

This property of cube roots stems from the fact that any real number has a unique real cube root. This is because the cube root function is the inverse of the cubing function, ${ x^3 }$ , which is defined for all real numbers.

Determining the Domain of f(x) = \sqrt[3]{5x + 25}

Now that we have a clear understanding of domain and cube root functions, we can tackle the problem of finding the domain of ${ f(x) = \sqrt[3]{5x + 25} }$ . The function involves a cube root, which, as we discussed, is defined for all real numbers. This is a crucial point.

The expression inside the cube root is ${ 5x + 25 }$ . Since we can take the cube root of any real number, the only potential issue would be if the expression ${ 5x + 25 }$ itself had any restrictions on its domain. However, ${ 5x + 25 }$ is a linear expression, and linear expressions are defined for all real numbers. There are no denominators, square roots, or logarithms within the expression that could cause any restrictions.

Therefore, we can conclude that the domain of ${ f(x) = \sqrt[3]{5x + 25} }$ is all real numbers. No matter what value we substitute for ${ x }$ , we will always get a valid real number output. This is a direct consequence of the properties of cube root functions and the fact that the expression inside the cube root is a simple linear function.

Mathematical Explanation

To further solidify our understanding, let's break down the mathematical reasoning:

The cube root function, ${ \sqrt[3]{u} }$ , is defined for all real numbers ${ u }$ .
In our function, ${ f(x) = \sqrt[3]{5x + 25} }$ , the expression inside the cube root is ${ u = 5x + 25 }$ .
The expression ${ 5x + 25 }$ is a linear function, which is defined for all real numbers ${ x }$ .
Therefore, the composition of the cube root function and the linear function, ${ f(x) = \sqrt[3]{5x + 25} }$ , is also defined for all real numbers ${ x }$ .

This step-by-step reasoning demonstrates why the domain of the function is all real numbers.

Representing the Domain

There are several ways to represent the domain of a function. Here are the most common methods for expressing the domain of ${ f(x) = \sqrt[3]{5x + 25} }$ :

Interval Notation

Interval notation is a concise way to represent a set of real numbers using intervals. The interval notation for all real numbers is ${ (-\infty, \infty) }$ . This notation indicates that the domain includes all numbers from negative infinity to positive infinity, excluding the infinities themselves (hence the parentheses).

Set Notation

Set notation uses set-builder notation to describe the elements of the domain. The set notation for all real numbers is ${ \{ x \mid x \in \mathbb{R} \} }$ . This is read as "the set of all ${ x }$ such that ${ x }$ is an element of the set of real numbers ${ \mathbb{R} }$ ."

Number Line

We can also represent the domain graphically on a number line. For all real numbers, we would shade the entire number line, indicating that every point on the line is included in the domain. We would use arrows at both ends of the line to signify that the domain extends infinitely in both directions.

Conclusion

In summary, the domain of the function ${ f(x) = \sqrt[3]{5x + 25} }$ is all real numbers, which can be represented in interval notation as ${ (-\infty, \infty) }$ or in set notation as ${ \{ x \mid x \in \mathbb{R} \} }$ . This is because cube root functions are defined for all real numbers, and the expression inside the cube root, ${ 5x + 25 }$ , is a linear function that is also defined for all real numbers.

Understanding how to determine the domain of a function is a crucial skill in mathematics. By carefully considering the types of operations involved (such as square roots, fractions, and logarithms) and their inherent restrictions, we can accurately identify the set of all possible input values for which the function is valid. In the case of cube root functions, the absence of restrictions makes them particularly straightforward to analyze in terms of their domain.

This comprehensive exploration of the domain of ${ f(x) = \sqrt[3]{5x + 25} }$ should provide you with a solid foundation for tackling similar problems in the future. Remember to always consider the properties of the functions involved and any potential restrictions they may impose on the domain.