another way to look at this is:
the inverse of a composition is the reverse composition of the inverses:
.
to see that this is so, recall that an inverse of a function (provided one does exist) is a function g so that f(g(x)) = g(f(x)) = x.
we can write this as:
.
so let's calculate
.
since
no matter what "t" is, we have (taking
),
.
the proof that
is entirely similar.
now let's look at "your function":
.
this is the composition of 3 functions:
, where
a little thought should convince you that:
(note that means k is g's inverse)
(h is its own inverse, this can happen)
(and as we would expect, g is k's inverse).
in short,
you might find this slightly amazing.
what this means is:
.
let's "try it" with some actual number, instead of x. how about x = 5?
ok, that's kind of a weird number. so let's find f(x) when
.
. huh. how about that?