Question

what is the inverse of the function f(x)=11sin(5x) and what is the domain to be a 1 to 1 function

Answer 1

inverse of trig functions is determined for some interval of the domain
so you should tell the interval for which you want to find the inverse.

Answer 2

but for the 2nd question
the period is 2pi/5
so!it's one to one over any interval of this period.

Answer 3

What is the formula for calculating the inverse, only given the function f(x)=11sin(5x)?

Answer 4

11sin(5x) is not a one to one function by itself(a horizontal line crosses it's graph more than once ...right?) so you can't deduce it's inverse. but if you are given some interval you say 2pi/5(which is it's period)
the inverse function would be
f(x)=5sin(inv.)(11x)...only about that period.