Problem 49 - Calculus Early Transcendentals

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 ).$$f(x, y, z)=\sqrt{y-z}.$$

Step-by-Step Solution

Answer

Answer: The domain of the function $$f(x, y, z)=\sqrt{y-z}$$ is $$D = \{ (x, y, z) : y \ge z \}$$.

Step 1: Establish the Condition for Functions Inside the Square Root

For a function defined with a square root, the term inside the square root must be non-negative. In this case, we have $$f(x, y, z)=\sqrt{y-z}$$, so we need to ensure that $$y-z$$ is non-negative. To satisfy this condition, $$(y-z) \ge 0$$ must be true.

Step 2: Solve the Inequality to Determine the Domain of the Function

Now we need to solve the inequality $$y-z \ge 0$$ to find out which values of $$y$$ and $$z$$ will make the function defined. The inequality is equivalent to $$y \ge z$$.

Step 3: Graph the Solution and Describe the Domain

Visualizing the inequality $$y \ge z$$ in a 3-dimensional ($$x, y, z$$) space, we can imagine that the domain consists of all points in the space where $$y$$ is greater than or equal to $$z$$, regardless of the $$x$$ value. Thus, the domain of the function $$f(x, y, z)=\sqrt{y-z}$$ is defined as $$D = \{ (x, y, z) : y \ge z \}$$. This is the set of all points in 3-dimensional space where the $$y$$ coordinate is greater than or equal to the $$z$$ coordinate, corresponding to a region above a diagonal plane with the equation $$y=z$$.

Key Concepts

Square Roots

Square roots are fascinating elements of mathematics. They allow us to find a number that, when multiplied by itself, equals a given number. In the context of functions, square roots have specific requirements to ensure a valid and real result. The expression inside a square root must be non-negative (greater than or equal to zero). This is because the square root of a negative number is not defined in the set of real numbers.

For a function like \( f(x, y, z) = \sqrt{y-z} \), it's essential to check the condition \( y - z \geq 0 \) to ensure the square root remains real and valid. The range of values that satisfy this inequality will provide us the domain of the function, ensuring that it will output real numbers under these conditions.

Inequalities

Inequalities are mathematical expressions used to determine the relative size or order of two values. They are crucial in defining the domain of functions involving square roots. When we encounter an inequality like \( y - z \geq 0 \), it sets a condition that must be true for the function to be well-defined.

Solving an inequality involves finding all possible values of the variables that make the inequality true. For our example, \( y \geq z \) is the solution. This means that for any point \((x, y, z)\), if the value of \( y \) is equal to or greater than \( z \), the expression inside the square root is non-negative. Inequations like this are not only applied in pure math but also in real-world problem-solving situations where conditions or constraints are present.

Three-Dimensional Space

Three-dimensional space adds depth to our understanding of functions involving multiple variables like \(x, y, z\). In 3D space, visualizing functions and their domains becomes more interactive and insightful.

When discussing the domain of \( f(x, y, z) = \sqrt{y-z} \), the inequality \( y \geq z \) describes a region above a plane in this space. Imagine the plane formed by the equation \( y = z \), which runs diagonally. The domain includes all points where \( y \) is greater than or equal to \( z \), covering everything above or on this plane. It paints a picture of a domain, somewhat like a thick layer, that spans the entire space in the direction where \( y \) exceeds \( z \).

This spatial perspective highlights how mathematical concepts can be visualized, making it easier to comprehend scenarios involving more than two dimensions.