SignificanceThe functions y = xn are power functions, so polynomials are made from power functions. More general exponents arise naturally in applications. For example, volume increases as the (3/2)th power of the surface area. "Root" functions such as the sqrt(x) and cube root of x are power functions.
Standard NotationPower functions are functions of the form f(x) = x2, f(x) = 4 x-3, f(x) = -9 x1/2, and so on. The general form is f(x)= kxp where p is any real number and k is non-zero.
Domains of Power FunctionsIf p is a non-zero integer, then the domain of the power function f(x) = kxp consists of all real numbers. For rational exponents p, xp is always defined for positive x, but we cannot extract an even root of a negative number. Thus x(1/4) is not defined for any negative real numbers. Neither is x(3/4) (the fourth root of x cubed).
Any rational number p can be written in the form p = r/s where all common factors of r and s have been cancelled. When this has been done, kxp has domain
Graphs of Power FunctionsThe shape of the graph of y = kxp depends on the sign of k and where p lies. The effect of the coefficient k is to scale and, if negative, flip the graph about the x-axis. All graphs of y = xp pass through the point (1,1). Beyond that, the shape and growth for large positive x is given by:
Calculations with Power FunctionsThe typical equation involving a power function is an equation of the form
kxp = b.
The solution to such an equation, if p solution exists, is given by
x = (b/k)(1/p).
as can be easily checked using the rules for exponentiation.