Take a look at the applet: Power Functions
Here you see the graphs of the power function at different values for . The function has the following properties for :
: the curve goes through points and and has an increasing (positive) slope.
: is a linear function through points en .
: the graph goes through points en and has a decreasing (positive) slope.
: the function is undefined for , the graph goes through point and has a decreasing (negative) slope, the -axis and the -axis are asymptotes of the graph.
For the function only exists if is a whole number is (or if is a fraction with an uneven denominator, such as , , , , etc). Depending on whether is positive or negative, the graph will be increasing or decreasing.
The equation has exactly one solution when , and as long as is not an even whole number (not ), because in that case there would be two solutions. The equation has exactly one solution when and if is an uneven whole number (not ).