# Power Function with Integer Negative Exponent

Consider function y=x^(-n), where n is natural number.

When n=1, we obtain that y=x^(-1) or y=1/x. This is hyperbola.

Let n is odd number greater than one: n=3, 5, 7, ... . In this case function y=x^(-n) has same properties as y=1/x. Graph of the function y=x^(-n) (n=3, 5, 7, ...) resembles graph of the function y=1/x (see left figure). If |x|<1 the the bigger n, the further graph from x-axis. If |x|>1 the the bigger n, the closer graph to x-axis.

Let n is even number, for example n=2.

Properties of the function y=x^(-2)=1/x^2 are following:

1. Function is defined for all x, except x=0.
2. Function is even.
3. Function is increasing on (0,+oo) and decreasing on (-oo,0).

Same properties have any functions of the form y=x^(-n), where n is even and greater than two (see right figure).