Power functions > Power functions
12345Power functions

Theory

Take a look at the applet: Power Functions

Here you see the graphs of the power function f ( x ) = x p at different values for p . The function has the following properties for x > 0 :

  • p > 1 : the curve goes through points ( 0 , 0 ) and ( 1 , 1 ) and has an increasing (positive) slope.

  • p = 1 : f is a linear function through points ( 0 , 0 ) en ( 1 , 1 ) .

  • 0 < p < 1 : the graph goes through points ( 0 , 0 ) en ( 1 , 1 ) and has a decreasing (positive) slope.

  • p < 0 : the function is undefined for x = 0 , the graph goes through point ( 1 , 1 ) and has a decreasing (negative) slope, the x -axis and the y -axis are asymptotes of the graph.

For x < 0 the function only exists if p is a whole number is (or if p is a fraction with an uneven denominator, such as 1 3 , 2 3 , 1 5 , 2 5 , etc). Depending on whether p is positive or negative, the graph will be increasing or decreasing.

The equation x p = a has exactly one solution when a > 0 , and as long as p is not an even whole number (not 0 ), because in that case there would be two solutions. The equation x p = a has exactly one solution when a < 0 and if p is an uneven whole number (not 0 ).

previous | next