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Chapter 1: Problem 23

Find the domain of each function. $$ \sqrt[5]{x} $$

Short Answer

Expert verified
Domain: \((-\infty, \infty)\).

Step by step solution

01

Understanding the Function

The function is given as \( \sqrt[5]{x} \), which is the fifth root of \( x \). This means for any real number \( x \), we need to have \( x \) such that \( \sqrt[5]{x} \) is a real number.
02

Identify Domain for Fifth Root

For odd roots, like the fifth root, we can take the root of any real number, whether positive or negative. There are no restrictions on the input value \( x \) because taking an odd root of a negative number still results in a real number.
03

Conclusion of the Domain

Since \( x \) can be any real number, the domain of \( \sqrt[5]{x} \) is all real numbers. We can express this in interval notation as \( (-\infty, \infty) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fifth Root
The fifth root of a number refers to a value that, when multiplied by itself five times, gives the original number. Unlike even roots such as square roots, odd roots like the fifth root can handle negative numbers. This means that you can calculate the fifth root of not just positive values, but negative and zero values too. If you remember nothing else, keep this in mind:
  • The fifth root of a positive number is positive.
  • The fifth root of zero is zero.
  • The fifth root of a negative number is still a real number, but it will also be negative.
This flexibility with negative numbers is what makes the fifth root applicable to all real numbers, allowing it to handle any real figure you throw at it.
Real Numbers
Real numbers are the vast and inclusive group of numbers that you frequently encounter in math. They include a variety of familiar types:
  • Positive numbers
  • Negative numbers
  • Zero
  • Decimals and fractions
However, they do not include imaginary numbers like the square root of (-1).
When determining the domain of functions like the fifth root, it's crucial to understand that all these elements come into play. Since the fifth root can take both negative and positive numbers, the domain of our function is all real numbers. This broad range is why the notion of real numbers is so integral in mathematics.
Interval Notation
Interval notation is a concise way of representing a set of numbers, showing which numbers are included in a set. In the case of the domain for the fifth root function, interval notation clearly expresses that all numbers from negative infinity to positive infinity are included.
  • The notation \((-\infty, \infty)\) indicates every real number is a part of this domain.
  • The parentheses imply that infinity is not a number that can actually be reached, but rather a concept that denotes an unbounded set.
Interval notation is helpful when we need to clarify the range of values that a function can take or be applied to. It simplifies the task of visualizing this range and keeps our mathematical expression both clear and precise.

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