Problem 7 - APEX Calculus

Question

In Exercises \(7-14,\) give the domain and range of the multivariable function. $$ f(x, y)=x^{2}+y^{2}+2 $$

Step-by-Step Solution

Answer

Domain: \( \mathbb{R}^2 \); Range: \([2, \infty)\).

Step 1: Identifying the Domain

The domain of a multivariable function like \( f(x, y) = x^2 + y^2 + 2 \) is the set of all possible pairs \((x, y)\) for which the function is defined. Since \( f \) is made up of squared terms and additions, which are defined for all real numbers, the domain of this function is all pairs \((x, y)\) in the real plane. Thus, the domain is \( \mathbb{R}^2 \) or \{ (x, y) | x, y \in \mathbb{R} \}.\".

Step 2: Deriving the Range

The range of a function is the set of possible output values. Since \( f(x, y) = x^2 + y^2 + 2 \) is a sum of non-negative terms \(x^2\) and \(y^2\), the smallest value for \( f(x, y) \) occurs when both \(x\) and \(y\) are zero. In this case, \( f(0, 0) = 0^2 + 0^2 + 2 = 2 \). As \(x^2 + y^2\) increases without bound as \(x\) and \(y\) move away from the origin, the range is all real numbers \( f(x, y) \geq 2 \). Therefore, the range is \([2, \infty)\).

Key Concepts

Domain of a Function

In mathematics, the domain of a function refers to the set of all possible input values that the function can accept. For a multivariable function like \( f(x, y) = x^2 + y^2 + 2 \), its domain consists of all the pairs \((x, y)\) for which the function is well-defined.

Understanding the domain is crucial because it establishes the foundation for where a function operates:

• Inspection of the expression reveals squared terms and constant addition, which do not impose restrictions on \(x\) and \(y\).
• The function \(f(x, y)\) accommodates any real number inputs since both squaring and addition are valid for all real numbers.

Therefore, the domain of the function \( f(x, y) = x^2 + y^2 + 2 \) is the entire set of real number pairs. Mathematically, this is expressed as \( \mathbb{R}^2 \) or \(\{(x, y) \mid x, y \in \mathbb{R}\}\), meaning every point in the two-dimensional real plane.

Range of a Function

The range of a function represents all possible output values it can produce based on its domain. For the multivariable function \( f(x, y) = x^2 + y^2 + 2 \), discerning its range involves analyzing the function's behavior as \((x, y)\) traverse their domain.

Key aspects to understand the range are:

• The terms \(x^2\) and \(y^2\) are always non-negative because squaring any real number yields zero or a positive result.
• Thus, the smallest possible output of \( f(x, y) \) happens when both \(x\) and \(y\) are zero, giving \(f(0, 0) = 0^2 + 0^2 + 2 = 2\).
• When \(x\) and \(y\) increase in magnitude, \(x^2 + y^2\) can approach any large value, increasing \(f(x, y)\) indefinitely.

Thus, the range of \( f(x, y) \) is continuous from the minimum value of 2 upwards, expressed as \([2, \infty)\).

Real Numbers

Real numbers form a fundamental concept in mathematics, encompassing all numbers that can be found on the continuous number line. They include both rational numbers (such as integers and fractions) and irrational numbers (such as \( \pi \) and \( \sqrt{2} \)).

Regarding functions, especially multivariable functions like \( f(x, y) = x^2 + y^2 + 2 \), understanding real numbers is essential because:

• They define the domain as all possible pairs \( (x, y) \) that include any real numbers.
• The real number system provides the basis for function operations like addition, multiplication, and squaring.

In essence, the set of real numbers \( \mathbb{R} \) allows for complete freedom in choosing values for \(x\) and \(y\) within the function’s domain, impacting both its domain and range calculations. In a multivariable context, \( \mathbb{R}^2 \) is the set of all real number pairs, describing the space where the function operates.