a mixed bag

Originally posted by BarefootChessPlayer
[b]first, `a who am i':
My first part is a bodily container reversed.
My second part is some of a toxic substance, backwards.
My last part is three vowels.
My whole is associated with the letter W.
Who am I?

next:
...[text shortened]... inverse is also a function, which maps onto a proper subset of itself.

no one even tried these???! i'm disappointed.

for the first one:
this is relatively obscure.
a bodily container (sac), reversed, is cas.
a toxic substance is poison, and we want part of it (pois), backwards: siop.
with that much, you can guess the vowels needed are e, i, and a.
all together, they spell Cassiopeia, a constellation depicting a (mythological?) queen of egypt, whose bright stars form a pattern resembling a W.

next:
this is one a friend gave me many years ago.
given the sequence 1, 2, 720, what is the next one, and give the rule?
the next is 24 factorial (24!), and the rule is: S(n) = ((n!)!) or the factorial of n factorial. since the factorial function in and of itself increases very rapidly, this one obviously gets very big very fast.

the third one is, as most of you have guessed, a collection of transposes, in this case, u. s. state capital cities.
thus:
TRACE MOANS = sacramento (california)
DIME NOSES = des moines (iowa)
A SOWN THING = washington (a state, or the capital city of the united states)
ERMINE PLOT = montpelier (vermont)
since washington is not a state capital, the one that doesn't belong is A SOWN THING.

the final one was an assignment in a math class i took in college. there are several possible answers. the one i came up with was the exponential function, e^x (exp(x)), which maps the entire set of real numbers onto the positive reals; and its inverse, the natural logarithm, log x or ln x (log(x) or ln(x)), which maps from the positive reals onto the entire set of reals. another is the arc tangent function, mapping the entire set of reals onto {x | -pi/2 < x < pi/2} (also written as the open interval (-pi/2, pi/2) if you prefer) and its inverse, the tangent of x for x in the above interval. still another is the hyperbolic tangent, tanh x (all reals onto (-1, 1) and its inverse, arc tanh x, which maps from (-1, 1) onto the entire set of reals. a fourth is on the integers. define the function `successor' as `the least integer after n', or n --> n + 1. if your domain is the positive integers (which i call Z+), the range is Z+ without the number 1. the inverse, which we will define as `predecessor', maps that set back onto Z+. so, possible answers are: exp(x) and log (ln) x, tan(x) over a specified interval and arctan(x), tanh(x) and arctanh(x), and the successor and predecessor functions.

now, don't you feel bad as not even attempting such simple questions?

Originally posted by kaushpaul
could you elaborate on the factoral answer...by substituting 2 to you solution of S(n) = ((n!)!) , ihgow do u get 720 as the next number?

(1!)! = 1
(2!)! = 2
(3!)! = 6! = 720

Originally posted by DumDum81
I'm confused😕

I will assume that when you say you are confused it is because you are unfamiliar with what the factoral function is. If that is true, then here is as simple an explination as I can give:

If you take the factoral of any number, you are essentially multiplying that number by every whole number below it. In other words, you are multiplying that number by one number less than it, then multiplying it again by one number less, and again..., until you have multiplied that number by every number below it to 1.

n! is the same thing as: n*(n-1)*(n-2)*(n-3)*...*1. So, if n equals 5, for example, then: 5! = 5*4*3*2*1 = 120.

(3!)! = (3*2*1)! = 6! = 6*5*4*3*2*1 = 720.

I hope that makes sense (assuming that was your question).

Originally posted by iamatiger
the sequence (2^n - 2) ! also starts with 1,2,720...

you're right, it does.
i never thought of that one!
the answer to that would be 14!, which is also quite large.
one thing i wanted to point out to all about factorials and computing them:
if you know how to program recursively, such as in "pascal", factorials are easy.
define:
0! = 1! =1
for any integer n > 1, n! = n*(n - 1)!
incidentally, the factorial function is continuous, and for non-integer values, its "partner", the gamma function, is used to get the fractional (probably irrational) part.