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/