When I wanted to graph $y=(-2)^x$ many graphing calculator apps refused to plot it. TI-Nspire CAS plotted it as shown in the first picture. I think the plot is not correct as only the envelopes should be there with no values between the envelopes as shown in the second picture and the $(-2)^x$ should look like $2^x$ and $-(2^x)$ plotted on the same graph but of course with many discontinuities as explained in my analysis below. Am I right?
Here’s my analysis of the function:
First: when x >=0
- if x is an integer, x>=0
(-2)^x oscillates back and forth When x is an even integer (-2)^x is positive. When x is an odd integer (-2)^x is negative.
- when x is a rational number, x>0 let x=p/q , p>0, q>0
if p is even and q is odd, (-2)^x is a positive real value Example: (-2)^(100/51)= 3.8927
if p is odd and q is odd, (-2)^x is a negative real value. Example: (-2)^(99/51)=-3.8402
if p is odd and q is even, (-2)^x is an imaginary value (not defined in the set of real numbers) Example: (-2)^(99/50)=i 3.9449
(Also in all the above cases, if you extend your analysis to include complex, we get q complex roots.)
In the domain of rational numbers, (-2)^x oscillate or is undefined (imaginary or complex)
- When x is irrational, x>0 (-2)^x has no real value. It has an infinite number of complex roots.
Second: when x<0 (-2)^x=1/(-2)^|x| Use the same approach above to analyze the behavior of the function.