Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain of each function. $$f(x)=\sqrt[5]{x+32}$$

Step-by-Step Solution

Answer

The domain of \( f(x) = \sqrt[5]{x+32} \) is all real numbers, \( x \in \mathbb{R} \).

Step 1: Understand the Problem

We are required to determine the domain of the function \( f(x) = \sqrt[5]{x+32} \). This involves finding all possible values of \( x \) for which the function is defined.

Step 2: Identify Restrictions for Fifth Roots

Consider the expression \( \sqrt[5]{x+32} \). Fifth roots are defined for all real numbers since any real number raised to the power of 5 results in a real number. This means there are no restrictions for the expression \( x+32 \).

Step 3: Define the Domain

Since \( x + 32 \) can be any real number for the fifth root to be defined, \( x \) can also be any real number. Therefore, the domain of the function is all real numbers.

Key Concepts

Fifth Roots

Fifth roots are an extension of the concept of square roots, but instead of finding a number that when squared gives us the original number, we look for a number that when raised to the power of five does so. This mathematical operation is quite flexible because any real number, positive or negative, has a fifth root.

• A positive number, when raised to the fifth power, yields a positive number.
• A negative number, when raised to the fifth power, yields a negative number.

This property makes fifth roots distinct from square roots, which are only defined for non-negative numbers in terms of real numbers. As such, when dealing with fifth roots, we do not have to worry about whether the expression inside the root is positive or negative. This attribute is crucial because it implies that fifth roots are defined for every real number.

Real Numbers

Real numbers are the foundation of many mathematical operations, including those involving roots and powers. They can be positive, negative, or zero, and include both rational numbers (like fractions and integers) and irrational numbers (like the square root of 2 or π).

• Real numbers are represented on the number line, covering everything from negative infinity to positive infinity.
• Operations with real numbers, including addition, subtraction, multiplication, division (except by zero), and root extraction are well-defined.

Knowing that a function's domain can often be all real numbers helps simplify the process of determining where the function is valid. For functions involving fifth roots, since every real number is acceptable, the domain is immediately understood as all real numbers without additional calculations.

Function Definition

Defining a function involves specifying a relationship between inputs from one set, called the domain, and outputs from another set, often the real numbers. In mathematical terms, if we have a function, say \( f \), and an input \( x \), the output \( f(x) \) is determined by a specific rule or formula associated with the function. The function given in the exercise, \( f(x) = \sqrt[5]{x+32} \), is an example of a function where the rule is taking the fifth root of \( x + 32 \). This definition of the function guides us in understanding how to map inputs to outputs in the function's domain.To define the domain correctly, we look at any restrictions that could arise from the mathematical operations involved. However, since fifth roots are well-defined for all real numbers and there are no divisions by zero or restrictions on \( x \) from the expression inside our fifth root, the domain is the whole set of real numbers.