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Re: Opposites

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1waffle1 is not online. 1waffle1
Joined: 16 Oct 2007
Total Posts: 16381
29 Oct 2011 02:22 PM
+ is the opposite of -
* is the opposite of /
x^y is the opposite of x^(1/y)

What's the opposite of a factorial? A factorial really isn't a mathematical function, it has limitations, you can only do it when the argument is a positive integer.
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kingkiller1000 is not online. kingkiller1000
Joined: 12 Dec 2008
Total Posts: 26415
29 Oct 2011 02:31 PM
Nothing. What can you think of that the opposite of a factorial could do?
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1waffle1 is not online. 1waffle1
Joined: 16 Oct 2007
Total Posts: 16381
29 Oct 2011 02:32 PM
720? = 6
No real purpose that I can think of.
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pwnedu46 is not online. pwnedu46
Joined: 23 May 2009
Total Posts: 7534
29 Oct 2011 02:35 PM
Since factorials are recursive multiplication (n! = n * (n-1) * (n-2) * ... * 1), the opposite of that would be recursive division. I don't think there's a specific name though.
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Aaaboy97 is not online. Aaaboy97
Joined: 05 Apr 2009
Total Posts: 6612
29 Oct 2011 02:35 PM
>x^y is the opposite of logₓ(y)

There is no defined opposite of a factorial, considering how you would need lookup tables to find the reverse.
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1waffle1 is not online. 1waffle1
Joined: 16 Oct 2007
Total Posts: 16381
29 Oct 2011 02:40 PM
(x^y)^(1/y) == x
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pwnedu46 is not online. pwnedu46
Joined: 23 May 2009
Total Posts: 7534
29 Oct 2011 02:40 PM
"x^y is the opposite of logₓ(y)"

No.
x^y = z is the opposite of logₓ(z)=y

ex:
3^5 = 243
log base 3 of 243 = 5
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Aaaboy97 is not online. Aaaboy97
Joined: 05 Apr 2009
Total Posts: 6612
29 Oct 2011 02:45 PM
oh oops, sorry about that

I've just barely began to learn about logarithms, and it's all from my own experimentation
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pwnedu46 is not online. pwnedu46
Joined: 23 May 2009
Total Posts: 7534
29 Oct 2011 02:48 PM
@aaa: Lol. Actually, I was about to post the same thing, but then I was lik wait...that's wrong. :3
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Varp is not online. Varp
Joined: 18 Nov 2009
Total Posts: 5333
29 Oct 2011 03:59 PM
They're called inverses. They're denoted as f^-1, if f is the function being inverted.

f^-1(f(x)) = x

Also, with the whole thing about exponents, the inverse of with a constant a:

f(x) = x + a
f^-1(x) = x - a

f(x) = x * a
f^-1(x) = x/a

And then, with exponents, it has two inverses, as it isn't commutative. You have the two functions:

f(x) = x^a
f^-1(x) = x^(1/a)

f(x) = a^x
f^-1(x) = log_a(x)

For the factorial function, you have the function:

f(x) = x * f(x-1)
f(1) = 1

While I'm not entirely sure how you would find a recurrence relation to express the inverse, you can just use the process of:

function inverseFactorial(x)
local r = 1
local v = 1
while v < x do
r = r + 1
v = v * r
end
if(v ~= x)then error(x .. " is not in the range of the factorial function.") end
return r
end

This is basically a trivial implementation of an inverse; it just uses the property of g(f(x)) = x and tests various f(n) until f(n) equals the argument passed to the inverse function. This process must always terminate since n! monotonically increases & has a discrete domain

(Also, you could look up the gamma function; while your answer then would be gamma^-1(x) which is... probably not easy to compute. It also only works on certain domains)
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NXTBoy is not online. NXTBoy
Joined: 25 Aug 2008
Total Posts: 4533
29 Oct 2011 04:43 PM
You might want to take a look at the [gamma function](http://en.wikipedia.org/wiki/Gamma_function)
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pwnedu46 is not online. pwnedu46
Joined: 23 May 2009
Total Posts: 7534
29 Oct 2011 05:53 PM
@NXT: uhh...wikipedia didn't help.
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