Problem 10 - Calculus Volume 3

Question

For the following exercises, find the domain of the function. $$f(x, y)=\frac{y+2}{x^{2}}$$

Step-by-Step Solution

Answer

The domain of \( f(x, y) \) is all \( (x, y) \) with \( x \neq 0 \).

Step 1: Identify Division Constraint

The function given is \( f(x, y) = \frac{y + 2}{x^2} \). For a function of the form \( \frac{a}{b} \), the denominator \( b \) cannot be zero because division by zero is undefined.

Step 2: Set Denominator Not Equal to Zero

To find where the function is undefined, set the denominator \( x^2 \) not equal to zero: \( x^2 eq 0 \).

Step 3: Solve the Denominator Equation

Solve the inequality \( x^2 eq 0 \). Since \( x^2 \) is zero only when \( x = 0 \), the restriction on \( x \) is \( x eq 0 \).

Step 4: Determine Domain

The domain of the function consists of all \((x, y)\) such that \( x eq 0 \). In mathematical terms, the domain is all real numbers for \( x \) and \( y \), except \( x = 0 \). So the domain is \( \{ (x, y) \mid x eq 0, \, y \text{ is any real number} \} \).

Key Concepts

Division Constraints

Understanding division constraints is crucial when dealing with functions like \( f(x, y) = \frac{y+2}{x^2} \). Whenever there is a division involved, we must ensure the denominator isn't zero. This is because division by zero is undefined in mathematics and would make the function meaningless or impossible to compute.

To identify the division constraint, examine the denominator portion of the function. In this case, the denominator is \( x^2 \). The constraint or condition we must enforce is that this denominator does not equal zero: \( x^2 eq 0 \). By establishing this, we ensure the function remains valid for all possible input values that meet this criterion.

Denominator Equation

The denominator equation is formed by the expression in the denominator of a function, and solving it is key to finding where the function might become undefined.

For the function \( f(x, y) = \frac{y+2}{x^2} \), the denominator is \( x^2 \). To solve the denominator equation, set \( x^2 = 0 \) and solve for \( x \).

• It tells us that \( x eq 0 \), as the only solution to \( x^2 = 0 \) is \( x = 0 \).

This equation helps us pinpoint which values could potentially cause division by zero, thus identifying them allows us to define the domain of the function correctly.

Real Numbers

The concept of real numbers is broad and essential in understanding the domain of functions. Real numbers include all the numbers that exist on the number line.

They include:

• Positive and negative integers (e.g., -3, 0, 42)
• Fractions and decimals (e.g., 1/2, 3.14)
• Irrational numbers (e.g., \( \sqrt{2} \), \( \pi \))

When defining the domain of \( f(x, y) = \frac{y+2}{x^2} \), we include all real numbers for both \( x \) and \( y \), except where division constraints occur. Thus, the domain is all pairs \((x, y)\) where \( x eq 0 \). This definition encompasses all real numbers for \( y \) and all real numbers for \( x \), with one important exception.

Mathematical Inequality

Mathematical inequalities play a crucial role in determining the constraints on the domain of a function. Inequalities allow us to express that one quantity is larger or smaller than another, rather than equal to.

In our function \( f(x, y) = \frac{y+2}{x^2} \), we use the inequality \( x^2 eq 0 \).

• This means \( x \) cannot be zero, ensuring the function remains valid.
• When writing the domain, this inequality stops \( x = 0 \) from being included.

This way, any real number other than zero is part of the domain for \( x \). Mathematical inequalities are fundamental when expressing restrictions and maintaining the integrity of functions in mathematical analysis.