91Ó°ÊÓ

The domain of a function encompasses all the input values (typically denoted as \( x \) and \( y \) for multivariable functions) where the function remains defined. For our example, the function \( f(x, y) = x^2 - y^2 \), there are no restrictions on \( x \) or \( y \). This means that both variables can be any real number because squaring any real number results in a valid real number output. Hence, the domain is all real number pairs \((x, y)\), denoted as \( \mathbb{R}^2 \).

It's important to understand that if a function has any mathematical expressions such as square roots, logarithms, or fractions, their domains could have restrictions—but here, we freely use all of \( \mathbb{R}^2 \).

The range of a function refers to all possible output values. For the function been analyzed, \( f(x, y) = x^2 - y^2 \), the potential output values are influenced by the squaring of both variables. Since \( x^2 \) ranges from 0 to infinity and \( y^2 \) ranges from 0 to infinity independently, depending on their values, \( x^2 - y^2 \) has the capacity to encompass all real numbers.

To see how the range could be all of \( \mathbb{R} \), consider:

• When \( x^2 > y^2 \), the result is positive.
• When \( x^2 < y^2 \), the result is negative.
• When \( x^2 = y^2 \), the result is zero.

Thus, no real number is off-limits for the function outputs, making the range \( \mathbb{R} \).

Level curves for a function are sets of points that yield the same output value, providing a contour map of constant function values. For \( f(x, y) = x^2 - y^2 \), a level curve is framed by \( x^2 - y^2 = c \), where \( c \) is a constant value. The equation \( x^2 - y^2 = c \) signifies a hyperbola.

Depending on the constant \( c \):

• If \( c > 0 \), the hyperbola opens along the \( x \)-axis.
• If \( c < 0 \), it opens along the \( y \)-axis.
• If \( c = 0 \), it forms the degenerate case where the level curve represents the lines \( x = y \) and \( x = -y \).

Level curves help visualize the behavior of the function as each curve represents an equation with equal \( f(x, y) \) values.

The concepts of open and closed sets touch on the inclusion of boundary points within the domain of a function. For the function \( f(x, y) = x^2 - y^2 \), with a domain \( \mathbb{R}^2 \), the understanding of open and closed becomes interesting.

A set is termed **open** if it does not include its boundary, and it's **closed** if it includes its boundary. In \( \mathbb{R}^2 \), which is essentially the entire plane, there is no real boundary, making it conventionally both open and closed. Because it extends infinitely without a defined "edge," there are no exterior points to consider.

• This unique property of being both open and closed is known as being "clopen."
• Often considered open since there isn’t a boundary to include or exclude."

This characterization is vital for understanding how the domain behaves in terms of topological properties.