91Ó°ÊÓ

Composite functions are functions formed by applying one function to the results of another. In the exercise, we explore the function \(f(x, y) = \cos \left(\frac{x y}{1 + x^2 + y^2}\right)\). This is a composite function because it combines a trigonometric function with a rational function.

The process of creating composite functions usually involves defining an inner function, then applying an outer function to it. In our case, the inner function is \(u(x, y) = \frac{x y}{1 + x^2 + y^2}\) and the outer function is \(\cos(u)\).

Composite functions are significant in many areas of mathematics because they allow the combination of simpler functions into a more complex one, while still analyzing the properties and behavior of each individual component.

Trigonometric functions are fundamental functions in mathematics, often related to angles. They include sine, cosine, and tangent, among others. These functions are periodic and remain continuous across their domains.

In the given exercise, we are mainly interested in the cosine function, \(\cos(u)\). This function exhibits the property that it is continuous for all real numbers.

For any input value, \(\cos\) smoothly maps the input to a range between -1 and 1. Because the cosine function does not have any discontinuities, if we compose it with any continuous function (as in our exercise with \(u(x, y)\)), the resulting composite function remains continuous wherever the inner function is continuous.

Rational functions are fractions where both the numerator and the denominator are polynomials. In this scenario, the function \(u(x, y) = \frac{x y}{1 + x^2 + y^2}\) is a rational function.

Rational functions are continuous on their domain, which includes all points where the denominator is not zero. For the function \(u(x, y)\), the denominator is \(1 + x^2 + y^2\), which is always greater than zero because both \(x^2\) and \(y^2\) are non-negative.

This ensures that \(u(x, y)\) is continuous everywhere in the plane \(\mathbb{R}^2\). Since the numerator and denominator form part of a polynomial function, and polynomials are continuous everywhere, this continuity naturally extends to rational functions over their valid domains.