91Ó°ÊÓ

In multivariable calculus, we extend the concepts of single-variable calculus, such as differentiation and integration, to functions of two or more variables. The domain of a multivariable function, like the one in the given exercise \( z = \frac{x+y}{xy} \) answers the question: 'For which pairs of \( x \) and \( y \) is this function defined?'. It is crucial to note that the function is undefined when either \( x \) or \( y \) is zero because division by zero is not allowed. Thus, the domain is all real numbers except for where \( x \) or \( y \) is zero. Multivariable functions can show a rich variety of behaviors compared to single-variable functions, primarily because there are more dimensions in which the function can change.

As we analyze such functions, we not only consider where they are defined but also explore their continuity, limits, and differentiability over their domain. Issues like finding maxima and minima, integration over regions, and understanding the geometry of functions become multi-dimensional puzzles, often requiring graphical or numerical methods alongside analytical ones to solve.

Rational functions are ratios of polynomials - in the simplest terms, one polynomial divided by another. The function in our exercise, \( z = \frac{x+y}{xy} \) is a rational function because the numerator \( x+y \) and the denominator \( xy \) are both polynomials. The domain of rational functions is all real numbers that do not make the denominator zero, for the reasons mentioned earlier: division by zero produces undefined results.

Rational functions can display various behaviors such as asymptotes, intercepts, and discontinuities. Notably, they often have vertical asymptotes where the denominator polynomials are zero, which leads to the function 'blowing up' to \( \infty \) or \( -\infty \) – exemplifying the function's behavior near points excluded from the domain. The range can be trickier to determine because it involves understanding the behavior and values that the function can reach, based on its algebraic structure and the constraints imposed by the domain.

Function analysis refers to the process of considering the properties of a function and its graph to understand its overall behavior. For functions like \( z = \frac{x+y}{xy} \) from the exercise, we want to analyze both the input, which leads us to the domain, and the output, which determines the range. After assessing that the domain excludes any pair of \( x \) and \( y \) that makes \( xy = 0 \) (since the function becomes undefined), we turn towards understanding the potential outputs, i.e., the range.

The range is found by looking at the possible values the function can take. For rational functions, we often find excluded values or intervals, particularly due to the function becoming undefined or due to asymptotic behavior. In the given case, the function approaches infinity as the product \( xy \) approaches zero, suggesting that the output can be very large or very small in magnitude, but never zero. Hence the range is all real numbers except zero. Function analysis encompasses not just the domain and range, but also the study of intervals where the function increases or decreases, convexity, and points of inflection, which provide an extensive understanding of the function's behavior.