2. \( f(x)=\frac{1}{x^{6}}= \)

This function, \( f(x) = \frac{1}{x^6} \), is an example of a rational function which showcases a power of negative integers. When graphing this function, you'll find it approaches zero as \( x \) moves away from zero in both the positive and negative direction, but it spikes to infinity as \( x \) approaches zero. It's fascinating to see how this behavior creates two distinct branches in the graph, one in the first quadrant and the other in the second! In real-world applications, functions of this form can model various phenomena, such as gravitational forces. For instance, the intensity of gravitational force decreases with the square of the distance from the center of the gravitational source, analogous to \( \frac{1}{r^n} \) where \( n \) is a positive integer. This principle helps us understand the movement of planets and even how we calculate forces in engineering and physics!