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.power function with negative integer exponent

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).