Problem 25 - APEX Calculus

Question

Give the domain and range of the functions of three variables. $$ f(x, y, z)=\sqrt{z-x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

Domain: \(z \geq x^2 - y^2\); Range: \([0, \infty)\).

Step 1: Understand the Function

The function is given by \( f(x, y, z) = \sqrt{z - x^2 + y^2} \). It includes a square root, which affects the domain due to the necessity that the expression inside must be non-negative.

Step 2: Determine the Domain Condition

For the expression under the square root to be valid, it must be non-negative. Thus, we need to find the condition where \( z - x^2 + y^2 \geq 0 \). This means \( z \geq x^2 - y^2 \).

Step 3: Express the Domain

The domain of the function consists of all points \((x, y, z)\) such that \( z \geq x^2 - y^2 \). There are no further restrictions on \(x\) and \(y\), so they can be any real numbers. Therefore, the domain is the set of all \((x, y, z)\) where this inequality is satisfied.

Step 4: Determine the Range of the Function

The range of a function is the set of all possible output values. Since the square root function produces non-negative values, the smallest value \(f(x, y, z)\) can take is 0 (when \(z = x^2 - y^2\)), and it increases without bound as \(z\) increases. Therefore, the range is \([0, \infty)\).

Key Concepts

Domain

When evaluating multivariable functions, the domain defines the set of all possible inputs for which the function is valid. In the given exercise, the function is \( f(x, y, z) = \sqrt{z - x^2 + y^2} \). A crucial feature of this function is the square root, which dictates certain conditions for its domain.

A square root function can only process non-negative numbers. Hence, the expression inside the square root, \( z - x^2 + y^2 \), must be greater than or equal to zero. This requirement translates to the inequality \( z \geq x^2 - y^2 \).

• The domain consists of all points \((x, y, z)\) where the inequality is satisfied.
• There are no restrictions on \(x\) and \(y\), meaning they can be any real number.

Understanding domains helps in visualizing and working with multivariable functions effectively by highlighting where the function can exist.

Range

The range of a function is about what outputs are possible from the inputs within the domain. For our function \( f(x, y, z) = \sqrt{z - x^2 + y^2} \), the output depends on the non-negative values created by the square root function.

As the square root of any real number is always non-negative, the smallest value our function can yield is 0. This happens when the expression inside the square root equals 0, specifically when \( z = x^2 - y^2 \).

• The range is all non-negative numbers, written as \([0, \infty)\).
• The function produces larger values as \(z\) grows larger than \(x^2 - y^2\).

Considering the range provides insight into the breadth of values a function can achieve, guiding the expectations of possible results.

Square Root Function

Square root functions are a foundational concept in both single-variable and multivariable calculus. They return only non-negative values (zero or positive), based on their characteristics.

For the given multivariable function \( f(x, y, z) = \sqrt{z - x^2 + y^2} \), the role of the square root is to ensure that the outputs remain non-negative if the expression inside it is valid. This makes it crucial to determine conditions like \( z \geq x^2 - y^2 \) for validity.

• Square roots tackle non-negative real numbers, protecting against imaginary results.
• They impose restrictions prompting inequalities on input expressions, crucial for the domain.

Recognizing the properties and limitations of square root functions aids significantly in the successful analysis of complex functional relationships.

Inequalities

Inequalities play a vital role in defining domains, especially in functions involving square roots. In mathematics, they encourage finding conditions under which expressions are valid. For \( f(x, y, z) = \sqrt{z - x^2 + y^2} \), we face the inequality \( z \geq x^2 - y^2 \).

Understanding and solving inequalities allows us to determine where a function makes sense. Solving \( z \geq x^2 - y^2 \) indicates that:

• The values of \(z\) must not be less than \(x^2 - y^2\) for real results of the function.
• It identifies a surface of inputs dictating a threshold for valid calculations.

Special attention to inequalities helps define the permissible input space, shaping domains and thus ensuring correct function evaluations.