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Chapter 2: Problem 117

What is the domain of the function \((3+x)^{1 / 4} ?\)

Short Answer

Expert verified
The domain of the function \(f(x) = (3+x)^{\frac{1}{4}}\) is all real numbers x such that \(x \geq -3\).

Step by step solution

01

Identify the Function and its Inner Expression

The given function is \(f(x) = (3+x)^{\frac{1}{4}}\). The expression inside the fourth root is \(3+x\).
02

Determine the Constraints on the Inner Expression

Since we are dealing with a fourth root, we know that the expression inside the root must be non-negative. That is, \(3+x \geq 0\).
03

Solve for the Domain

Now, we will solve the inequality \(3+x \geq 0\) to determine the valid values of x. 1. Subtract 3 from both sides of the inequality: \(x \geq -3\) So, the domain of the function \(f(x) = (3+x)^{\frac{1}{4}}\) includes all real numbers x such that \(x \geq -3\).

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical expressions that describe the relative size or order of two values. They are essential when defining domains for functions, especially root functions. In the given problem, we have the inequality \(3+x \geq 0\). This tells us that the expression inside the fourth root must be non-negative, meaning it can't be less than zero.

Here's how inequalities work in general:
  • \(a > b\): \(a\) is greater than \(b\)
  • \(a < b\): \(a\) is less than \(b\)
  • \(a \geq b\): \(a\) is greater than or equal to \(b\)
  • \(a \leq b\): \(a\) is less than or equal to \(b\)


For our function, finding the domain requires solving the inequality \(3+x \geq 0\) which simplifies to \(x \geq -3\). This means that the function is defined for any \(x\) value greater than or equal to \(-3\).
Exploring Real Numbers
Real numbers encompass all numbers that can be found on the number line. This includes both rational and irrational numbers. When defining the domain of a function, it's essential to consider which real numbers the function can accept.

Here's a quick breakdown of real numbers:
  • Rational Numbers: Numbers that can be expressed as a fraction, such as \(\frac{1}{2}\) or 3.
  • Irrational Numbers: Numbers that cannot be expressed as fractions, such as \(\pi\) or \(\sqrt{2}\).
  • Integers: Whole numbers, including negatives, zero, and positives like -2, 0, 1.
  • Whole Numbers: Non-negative integers, like 0, 1, 2.


The domain \(x \geq -3\) includes all these types of numbers. So, any real number greater than or equal to \(-3\) is a valid input for the function \(f(x) = (3+x)^{1/4}\).
Demystifying Root Functions
Root functions, like the fourth root in our exercise, involve finding values that, when raised to a specific power, equal the original expression. The function \(f(x) = (3+x)^{1/4}\) is defined as the principal fourth root of \(3+x\).

Understanding the nature of roots is crucial:
  • The square root \(\sqrt{x}\) is the most common, but any \(n\)th root can be found.
  • Only non-negative numbers can have real-valued even roots, like the fourth root.
  • Negative numbers can have real odd roots, but not even ones.


For this function, ensuring \(3+x\) is non-negative allows us to find a real fourth root. Thus, the inequality \(3+x \geq 0\) leads directly to the domain condition \(x \geq -3\). This non-negative requirement ensures the root function gives a real result.

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Most popular questions from this chapter

Suppose \(x, y,\) and \(z\) are real numbers and \(m\) is a positive integer. Explain why $$ x^{m} y^{m} z^{m}=(x y z)^{m} $$

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