I have seen several references that said I cannot have an exponential function in which the base is less than zero. When I saw that I thought to myself, “Oh yeah, well **** u. Watch this m** ******.” Then I came up with the function f(x) = (-2) to the x power where x is a real number (but not every real number as discussed below). If you say I can’t call my function an exponential function that’s OK. It’s just a name. But it’s still a function, right? The domain of f(x) is interesting:
x must be a rational number, and
any x that can be expressed as a ratio with an odd numerator and an even denominator is excluded from the domain.
Thus, f(x) is not continuous anywhere in its domain. A “graph” of f(x), if you could graph it, appears to branch off in two directions as x gets larger. One branch moves towards positive infinity and the other brach moves toward negative infinity. The value of f(x) jumps back and forth between the positive branch and the negative branch as x increases.
I have three questions for the members of this board:
(1) Is my description of f(x) correct?
(2) How do you make a superscript in the thread editor so I can type (-2)x instead of (-2) to the x power?
(3) Should I give up on amateur math as a hobby and go back to birdwatching?
The problem with this is that (-2)^(1/n) is often defined as the positive solution to the equation x^n = -2. If there is no such solution, the above term is undefined as well. One reason for this is to rule out weird functions like the one you are proposing.
Use [ sup ] and [ /sup ] tags (without spaces).
Nope, you aren’t actually wrong about anything and show a good understanding of the math involved. And being wrong isn’t bad either as long as you accept corrections.
Ways of showing exponentials, as used by people using computers:
(-2)**r
(-2)^r
pow(-2,r)
Pow is from c, which, didn’t include much in the way of native math operations.
From your description, y=(-2)**(1/2) is an example of one point of your discontinuous function. This has two (complex) possible values
+j sqrt(2)
-j sqrt(2)
so y is not a “function”. of x over that range.
(-2)[sup].5[/sup] can be solved by typing (-2)^.5 into the input window at https://www.wolframalpha.com. It has a complex-value solution, just not a real-value solution.
One way to find out how to write something like a[sup]b[sup]c[sup]d[sup]e[/sup][/sup][/sup][/sup] is to Quote and reply-to the post and see the coding. (You can start the Reply, but abandoื ระ before clicking Submit.)
I intentionally restricted the domain of f(x) to real numbers. That is where the domain restriction “any x that can be expressed as a ratio with an odd numerator and an even denominator is excluded from the domain” comes from. There is a complex number that satisfies the function at each of these points.
After a cup of coffee, it looks like there are two complex numbers that satisfy the function at each of these points. Then it wouldn’t even be a function anymore.
There is indeed a real valued function that is never continuous. But that aspect is enough to make it something not worth dealing with in most circumstances, hence the usual restrictions.
Those real valued answers still show up when you use the complex definition of exponentiation and logarithm, making it more convenient to only deal with them there, as a special a + 0i exception.
And, if you want convenient buttons for superscript, you should install my S-button Adder userscript. Though I note that superscript and subscript are only available on the Reply or “Go Advanced” screen, copying their location from other message boards. You can also install the SDMB Editor shortcuts if you want keyboard shortcuts–though they still don’t work in the inline Edit Post boxes.