Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe geometrically the domain of each of the indicated functions of three variables. \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-16}\)

Step-by-Step Solution

Answer

The domain is the set of all points outside or on the surface of a sphere with radius 4 centered at the origin.

Step 1: Understanding the Function

The function is given as \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16} \). It is a function of three variables \(x\), \(y\), and \(z\) with a square root.

Step 2: Identifying the Condition for the Domain

Since the function involves a square root, the expression inside the square root must be non-negative for the function to be defined. Mathematically, this condition is \( x^2 + y^2 + z^2 - 16 \geq 0 \).

Step 3: Solving the Domain Inequality

Rearranging the inequality, we arrive at \( x^2 + y^2 + z^2 \geq 16 \). This can be interpreted geometrically as all points \((x, y, z)\) that lie outside or on the surface of a sphere.

Step 4: Describing the Geometrical Domain

The inequality \( x^2 + y^2 + z^2 \geq 16 \) defines the region outside a sphere of radius 4, centered at the origin (0, 0, 0). Thus, the domain includes all points outside or on this sphere.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible inputs (or values of variables) for which the function is defined. In simpler terms, it tells us where our function "works" and what values we can plug into it without breaking any mathematical rules. In the case of the function \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16} \), we're interested in knowing for what combinations of \(x\), \(y\), and \(z\) this function will make sense.

Since this function involves a square root, the expression inside the root must be zero or positive because the square root of a negative number isn't defined within the realm of real numbers. This criterion gives us the inequality \( x^2 + y^2 + z^2 - 16 \geq 0 \). In solving the inequality, we find that the valid domain is any point \((x, y, z)\) where the sum of the squares of \(x\), \(y\), and \(z\) is at least 16.

Understanding this helps to determine where the function exists and guides us on proper and valid inputs when dealing with multivariable functions. These conditions keep our calculations within the realm of real numbers, giving our function a real and meaningful outcome.

Geometric Interpretation

Geometrically interpreting the domain of multivariable functions helps us visualize the problem in space. For our function \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16} \), the inequality \( x^2 + y^2 + z^2 \geq 16 \) describes the space outside or on a sphere in three-dimensional space.

Here's how we visualize this:

• The equation \( x^2 + y^2 + z^2 = 16 \) represents a sphere.
• The center of the sphere is at the origin, which is the point (0, 0, 0).
• The radius of the sphere is \( \sqrt{16} = 4 \).

This means any point \((x, y, z)\) that lies on or outside this sphere is included in the domain. Inside the sphere, the function is not defined because it would result in a negative value inside the square root. Visualizing the domain in this way gives us a better grasp of how the variables interact and the limitations of the function.

Seeing the domain as a tangible area around a sphere illustrates the limitations and possibilities of the function in the three-dimensional space, making the math more intuitive.

Inequality Solutions

Solving inequalities is a crucial step in determining the domain of functions like \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16} \). This process reveals which values of our variables result in a real number output, thus identifying the domain of our function.

Let's look at inequality \( x^2 + y^2 + z^2 \geq 16 \). To solve this:

• This is a quadratic inequality which resembles a circle's equation in 3D, extended to a sphere.
• Checking values: For instance, if \( x = 4 \), \( y = 0 \), and \( z = 0 \), then \( 4^2 + 0^2 + 0^2 = 16 \), which satisfies the inequality.
• For points further from \( (0, 0, 0) \), such as \( (4, 2, 2) \), \( 4^2 + 2^2 + 2^2 = 24 > 16 \), also satisfies the condition.

This exploration into inequality solutions shows us that the domain includes all points satisfying \( x^2 + y^2 + z^2 \geq 16 \), providing a clear pathway to understand where our function is usable and defined. Practicing these steps allows a student to handle and interpret functions involving inequalities with confidence.