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Chapter 8: Problem 405

Explain how to find the domain of a fourth root function.

Short Answer

Expert verified
The domain of a fourth root function \[f(x) = \sqrt[4]{x}\] is \[ [0, \infty) \].

Step by step solution

01

Understand the Fourth Root Function

A fourth root function is of the form \[f(x) = \sqrt[4]{x}\]. The value inside the root, \(x\), must be non-negative because the fourth root of a negative number is not a real number.
02

Identify the Domain Restriction

To determine the domain, set the expression inside the fourth root greater than or equal to zero. This gives the inequality \[x \geq 0\]. This means \(x\) must be non-negative for the function to output real numbers.
03

Write the Domain in Interval Notation

Based on the inequality, the domain of the function is all non-negative real numbers. In interval notation, this is written as \[ [0, \infty) \].

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Key Concepts

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

Domain Restrictions
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For a fourth root function, such as \( f(x) = \sqrt[4]{x} \), determining the domain involves finding the x-values that make the expression inside the fourth root defined.
Since taking the fourth root of a negative number does not result in a real number, the value inside the fourth root, x, must be non-negative.
In other words, it must be greater than or equal to zero.
This restriction ensures that the function outputs real numbers.
Mathematically, you can write this domain restriction as the inequality: \( x \geq 0 \).
Interval Notation
Once you have identified the domain restriction, you can express the domain in interval notation.
Interval notation is a concise way of representing ranges of values.
For the fourth root function, the domain is all x-values that are non-negative, meaning x can be zero or any positive value.
In interval notation, this is written as \( [0, \infty) \).
The square bracket [ means that the value 0 is included in the domain, while the parenthesis ) means that the interval extends infinitely in the positive direction.
So, \( [0, \infty) \) represents all real numbers from 0 to positive infinity, including 0 but not infinity.
Real Numbers
To fully understand the domain of the fourth root function, it's important to know what real numbers are.
Real numbers include all the numbers on the number line.
This includes all the positive numbers, zero, and all the negative numbers.
However, in the case of a fourth root function, we focus on non-negative real numbers due to the requirement that the value inside the fourth root must not be negative for the function to produce real outputs.
Real numbers are denoted by the symbol \( \mathbb{R} \).
In the context of our domain restriction, we're specifically interested in non-negative real numbers, which are indicated by \( [0, \infty) \) in interval notation.

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