Graph of y = (-n)^x
Date: 01/17/2005 at 23:19:51 From: Alexander Subject: A negative number to the power of x I am curious as to what the graph of y = (-n)^(x) would look like, such as y = (-2)^x. My graphing calculator will not show the graph as anything, but has many real values in a table of values. It is confusing becuase a negative number to the power of let's say 0.6 (3/5, so odd root) is a real number while a negative number to the power of 0.7 (7/10, so even root) would be an imaginary number. This would mean there are many random points that are both real and imaginary on a graph of this sort. Is it possible to construct a graph of y = (-n)^x becuase of this interesting coincidence that happens with odd and even roots of the negative numbers?
Date: 01/18/2005 at 09:48:45 From: Doctor Vogler Subject: Re: A negative number to the power of x Hi Alexander, Thanks for writing to Dr. Math. It depends partly on what you mean by "(-n)^x". You see, generally exponents are either defined algebraically for rational exponents or continuously (through calculus) for positive bases. If you have a negative base and want all exponents (including irrationals) then things get ugly. See also Base of an Exponential Function http://mathforum.org/library/drmath/view/55604.html Notice that in the case of all irrational and many rational exponents, the only possible value for y is complex. When graphing, you generally don't graph complex solutions. (Did you plot (i+1, 2i) on your graph of y = x^2? I didn't think so.) So your graph will essentially be dotted lines where the rational values take you. If you plotted those values where x is rational, then you would have a dotted line along y = n^x (since y = n^x when x has an even numerator and odd denominator) and a dotted line along y = -(n^x) (since -y = n^x when x has an odd numerator and odd denominator). If you want to consider what are the complex values of y = (-n)^x, then you have a completely different beast, since there are more than one possible values for y unless x is an integer, and there are infinitely many such values unless x is rational. See also Complex Powers http://mathforum.org/library/drmath/view/60383.html If you have any questions about this or need more help, please write back, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
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