Problem 37 - Calculus 3

Question

Find the domain of the following functions. $$ f(x, y, z)=\sqrt[3]{16-x^{2}-y^{2}-z^{2}} $$

Step-by-Step Solution

Answer

The domain is all real numbers \((\mathbb{R}^3)\).

Step 1: Understand the Problem

We are asked to find the domain of the function \( f(x, y, z) = \sqrt[3]{16 - x^2 - y^2 - z^2} \). The domain consists of all the input values \((x, y, z)\) for which the function is defined.

Step 2: Identify Function Type

We observe that the function includes a cube root. Cube roots, unlike square roots, are defined for all real numbers. Thus, the expression under the cube root, \(16 - x^2 - y^2 - z^2\), can be any real number.

Step 3: Determine Expression Constraint

For the function \(f(x, y, z)\), any real number can be cube rooted, which means \(16 - x^2 - y^2 - z^2\) can be any real number. There's no restriction on the values \(x, y, z\) can take based on making the cube root undefined; however, large values of \(x^2 + y^2 + z^2\) could make the function negative under cube root but it is still defined.

Step 4: Analyze and Determine Domain

Since there are no restrictions from the cube root, the values of \(x, y, z\) can range over all real numbers. Therefore, the domain of \( f(x, y, z) \) is all real number triples \((x, y, z)\).

Step 5: State the Domain

The function \(f(x, y, z)\) is defined for all \((x, y, z)\in \mathbb{R}^3\), meaning any real number can be used for \(x\), \(y\), and \(z\). So, the domain of \(f(x, y, z)\) is \(\mathbb{R}^3\).

Key Concepts

Domain of a Function

When we talk about the domain of a function in multiple variables, we're considering all possible combinations of input values that keep the function defined. In the case of the function \( f(x, y, z) = \sqrt[3]{16 - x^2 - y^2 - z^2} \), we need to ensure that the expression under the cube root, \( 16 - x^2 - y^2 - z^2 \), can take on any real value.

The domain of the function is directly connected to its ability to operate without violating any mathematical rules. For functions involving roots, such as cube roots, we must ensure the expression under the root is defined and permissible for the operation.

Remember these key points about domains:

• Determine if the mathematical operation constrains the input values.
• Cuberoots, unlike squareroots, are more lenient with what inputs are allowed because they can handle negative numbers as well.
• Here, the domain simplifies to all real numbers \((x, y, z)\).

Hence, for our function, the domain is all real numbers or \( \mathbb{R}^3 \), meaning every real number can potentially be used as an input for \(x\), \(y\), and \(z\).

Cube Root

The cube root concept differs from the more familiar square roots in a few key ways. When you find the cube root of a number, you're looking for a value that, when multiplied by itself twice, gives you the original number.

This mathematical operation can be applied to any real number—positive, negative, or zero. For example:

• The cube root of \(8\) is \(2\), because \(2 \times 2 \times 2 = 8\).
• The cube root of \(-8\) is \(-2\), since \((-2) \times (-2) \times (-2) = -8\).

Unlike square roots, which only deal with non-negative numbers to remain within the set of real numbers, cube roots extend into the negative realm without issue. This is crucial because it means in our function \( f(x, y, z) = \sqrt[3]{16 - x^2 - y^2 - z^2} \), regardless of the value that \(16 - x^2 - y^2 - z^2\) assumes (positive, negative, zero), the cube root will always be defined.

Thus, for functions involving cube roots, there's a broad acceptance of input values, making it simpler to determine the domain.

R^3

The notation \( \mathbb{R}^3 \) represents the set of all possible ordered triples of real numbers, namely \((x, y, z)\). Essentially, it's a three-dimensional space where every point is determined by a set of three numbers, each representing a position along the x-axis, y-axis, and z-axis.

In the context of the function \( f(x, y, z) \), saying its domain is \( \mathbb{R}^3 \) implies that all combinations of real numbers for \(x\), \(y\), and \(z\) are acceptable inputs. The space \( \mathbb{R}^3 \) encompasses every conceivable value for these variables without any constraints or limitations imposed by the function's structure.

• Each coordinate corresponds to one component in the triple \((x, y, z)\).
• The values can be spread across negative and positive infinity in each direction.
• No restrictions from cube roots mean each possible point in \( \mathbb{R}^3 \) fully describes where \(f\) is defined.

Understanding \( \mathbb{R}^3 \) helps clarify the domain of multivariable functions, highlighting its significance in describing spaces in mathematics, physics, and engineering.