91Ó°ÊÓ

Undefined values are those inputs to a function that result in an expression that is not mathematically valid. This often occurs due to certain operations that cannot be completed, like dividing by zero. In the context of the function \( f(x, y) = \frac{y}{x} - \frac{x}{y} \), we have to look closely at the parts of the function where division is involved. If we plug in a value for \( x \) or \( y \) that makes the denominator zero, the function becomes undefined. Specifically, this happens when \( x = 0 \) or \( y = 0 \). Such inputs are not acceptable because they do not produce valid, finite results. Undefined values are critical to identify because they help us in determining which values are valid inputs for our function.

Set notation is a mathematical way of expressing collections of objects or numbers, often used when defining domains for functions. For example, in the exercise given, the domain of the function \( f(x, y) = \frac{y}{x} - \frac{x}{y} \) is determined by the constraints \( x eq 0 \) and \( y eq 0 \). In set notation, we express this domain as follows:

• First, we specify that our ordered pairs \((x, y)\) belong to \( \mathbb{R}^2 \) (the set of all real number pairs).
• Next, we include the conditions \( x eq 0 \) and \( y eq 0 \) to show the restriction on the domain.

Thus, the domain in set notation is: \[ D = \{ (x, y) \in \mathbb{R}^2 \mid x eq 0, y eq 0 \} \] This notation effectively communicates the valid input values for the function, ensuring clarity and accuracy.

Division by zero is one of the fundamental situations that leads to undefined values in mathematical functions. It occurs when a number is divided by zero, which is not possible within the standard rules of arithmetic because there is no number that multiplied by zero returns the numerator.
In the function \( f(x, y) = \frac{y}{x} - \frac{x}{y} \), examining each fraction separately:

• For \( \frac{y}{x} \), if \( x = 0 \), this results in an undefined situation as you are dividing by zero.
• Similarly, for \( \frac{x}{y} \), if \( y = 0 \), it is also undefined due to division by zero.

Thus, ensuring that neither \( x \) nor \( y \) is zero avoids the undefined behavior caused by division by zero. Understanding this concept is crucial when defining the domain of a function because it prevents errors in mathematical operations and maintains the integrity of the function.