91Ó°ÊÓ

Understanding the **domain of a function** is crucial in the study of multivariable functions. The domain refers to all the possible input values (in this context, pairs of numbers) for which the function is defined. In other words, it’s the set of x-values and y-values that you can plug into the function without running into mathematical issues.

For the function \( g(x, y) = \frac{1}{xy} \), we need to make sure that the expression \( xy \) in the denominator doesn't equal zero, because division by zero is undefined. Therefore, both \( x \) and \( y \) must be non-zero. The solution to this is to exclude any points where \( x = 0 \) and \( y = 0 \).

In more formal terms:

• The domain of \( g \) is every pair of real numbers \((x, y)\) except the point \((0, 0)\).
• This is mathematically represented as: \( D(g) = (x,y) \in \mathbb{R}^2 : (x,y) eq (0,0) \).

Knowing this helps avoid situations where the function could "blow up" or be undefined.

The **range of a function** is the set of all possible output values (or z-values) that a function can produce. For the function \( g(x, y) = \frac{1}{xy} \), the task is to determine the possible values it can output when given various inputs within its domain.

Key points to remember:

• Any non-zero number divided by another non-zero number results in a non-zero value.
• Since \( 1 \) is always the numerator in \( g(x, y) \), the output will also never be zero.

Therefore, in mathematical terms, the range is all real numbers except zero, represented as \( R(g) = z \in \mathbb{R} : z eq 0 \). This means that the function \( g \) will output any real number, but it can never output zero.

**Division by zero** is one of the most critical concepts to grasp in mathematics. When you divide any number by zero, it leads to an undefined result. This is because dividing by zero does not provide a finite number or a result that makes sense mathematically.

Here’s why division by zero is problematic:

• Any number divided by zero would imply there is a number which, when multiplied by zero, gives back the original number, which is impossible as anything times zero is zero.
• A function with a zero in the denominator might misbehave and become unbounded or even stop being defined at that point.

For the function \( g(x, y) = \frac{1}{xy} \), we avoid zero in the denominator by not allowing either \( x \) or \( y \) to be zero. Thus, division by zero is prevented, ensuring that \( g \) remains a well-defined function throughout its domain.