Power FunctionsSignificanceThe functions y = x^{n} 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) = x^{2}, f(x) = 4 x^{3}, f(x) = 9 x^{1/2}, and so on. The general form is f(x)= kx^{p} where p is any real number and k is nonzero.Rules of exponentiationThe Rules of exponentiation are:
Domains of Power FunctionsIf p is a nonzero integer, then the domain of the power function f(x) = kx^{p} consists of all real numbers. For rational exponents p, x^{p} 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, kx^{p} has domain
Graphs of Power FunctionsThe shape of the graph of y = kx^{p} 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 xaxis. All graphs of y = x^{p} 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 formkx^{p} = 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.
