Advanced Math/Finding natural domain
Hello, I have a question about f(x)=x^pi.
Why the natural domain is 0 to infinity? why negative values are not included in it?
Negative numbers raised to a power can be tricky. One sign that things are different than for positive numbers is to consider x^1/2 when x <0. This is an imaginary number, i.e., proportional to i = (-1)^1/2. Also, the expression log(x^r) = rlog(x) is valid for x>0 but is undefined for x<0.
Negative numbers can be raised to a power for an (infinite) class of real number exponents. For instance, consider -1 raised to an integer, such as (-1)^3 = (-1)(-1)(-1) = -1. But note that (-1)^2 = +1. This bouncing between +/- 1 continues into the rational numbers, i.e., ratio of integers. Consider the two rational numbers r1 = 101/101 and r2 = 102/101; they are very close in value but (-1)^r1 = -1 while (-1)^r2 = +1.
This behavior has consequences when we consider irrational numbers such as pi. For real numbers x (the base) raised to an exponent r, the exponentiation x^r for r = irrational can be determined, or at least shown to exist, by taking the limit of the rational numbers that bracket it as they converge to r. However, for negative bases x, the oscillating nature of of x^r as decribed above means that x^r is discontinuous near r and the limit cannot be taken as with positive bases. Thus x^r cannot be defined for r irrational. For this reason, the domain of x must be confined to positive numbers for irrational exponents.