Question

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). $$p(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-9}$$

Short answer

Answer: The domain of the function $$p(x, y, z) = \sqrt{x^{2}+y^{2}+z^{2}-9}$$ consists of all points (x, y, z) on or outside a sphere of radius 3 centered at the origin, described by the inequality $$x^{2}+y^{2}+z^{2} \ge 9$$.

Step-by-step solution

Set up the inequality

Since the function p(x, y, z) is defined only when the expression inside the square root is non-negative, we must solve the inequality: $$x^{2}+y^{2}+z^{2}-9 \ge 0$$

Solve the inequality

Add 9 to both sides of the inequality to isolate the sum of squares: $$x^{2}+y^{2}+z^{2} \ge 9$$

Interpret the inequality geometrically

The left side of the inequality represents the sum of the squares of the Euclidean distances from the origin (0, 0, 0) to the point (x, y, z). The right side is a constant (9) which is the square of the radius (3) of a sphere centered at the origin. The inequality tells us that the distance from the origin to the point (x, y, z) must be greater than or equal to 3 units, which describes all points on or outside a sphere of radius 3 centered at the origin.

Write the domain of the function

The domain of the function $$p(x, y, z) = \sqrt{x^{2}+y^{2}+z^{2}-9}$$ consists of all points (x, y, z) on or outside a sphere of radius 3 centered at the origin, in the form of the inequality: $$x^{2}+y^{2}+z^{2} \ge 9$$

Key concepts

Domain of a Function

Understanding the domain of a function, especially a multivariable one like \( p(x, y, z) = \sqrt{x^{2}+y^{2}+z^{2}-9} \), is crucial in mathematics. The domain includes all the possible input values that won't cause the function to output an error or undefined value. For square root functions, the domain is determined by the condition that the radicand (the expression inside the square root) must be non-negative. This is because the square root of a negative number is not a real number. Consequently, when identifying the domain of \( p(x, y, z) \), one should solve an inequality to ensure the expression under the square root is greater than or equal to zero.

Inequality Solving

Inequality solving is a foundational skill required to find the domain of functions involving square roots. For the given function \( p(x, y, z) \), the inequality \( x^{2}+y^{2}+z^{2}-9 \ge 0 \) needs to be solved. The solution process often starts with isolating the variable or sum of variables, as seen in the step-by-step solution where the sum of squares is isolated on one side. By adding 9 to both sides, we determine a condition that the variables must satisfy. The resulting inequality, \( x^{2}+y^{2}+z^{2} \ge 9 \), indicates that any permissible values for \(x\), \(y\), and \(z\) must ensure that their sum of squares is greater than or equal to 9. Standard algebraic techniques, such as adding the same value to both sides and factoring, are key tools to solve these inequalities.

Euclidean Distance

The Euclidean distance is the straight-line distance between two points in Euclidean space. In our exercise, the Euclidean distance is between the origin (0, 0, 0) and any point \( (x, y, z) \) in three-dimensional space. Mathematically, it's represented by the expression \( \sqrt{x^{2}+y^{2}+z^{2}} \), which matches the radicand of our function \( p(x, y, z) \). The square of the Euclidean distance, hence, is \( x^{2}+y^{2}+z^{2} \), which must be greater than or equal to the square of the sphere's radius as per the inequality we are considering. When working with Euclidean distances, such as finding the domain of a function that includes them, we directly apply the concept of distance in our calculations. Here, it serves to define a solid geometric boundary—the sphere.

Geometric Interpretation

A geometric interpretation of an inequality can provide a clearer, more visual understanding of a function's domain. In our example, \( x^{2}+y^{2}+z^{2} \ge 9 \) describes a spherical region in three-dimensional space. Specifically, when visualized, it corresponds to all the points that lie on the surface of or outside a sphere with a radius of 3 units centered at the origin. This visual representation helps to comprehend the 'shape' of a domain, particularly for multivariable functions like \( p(x, y, z) \), and is a valuable tool for studies in fields such as calculus and analytic geometry. It effectively translates the algebraic condition of our function's domain into a tangible and more accessible spatial scenario.