The answer is already given in your question. The inverse of a function f is f-1, g is g-1, h is h-1 and i is i-1.
To explain why the inverse of <span>f = {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)} is f-1 = </span><span>f -1 = {(2,1), (3,2), (4,3), (5,4), (6,5), (7,6)} and that of the other functions to their corresponding inverses is because of the definition of the inverse of a function which is f = (x, y) f-1 = (y, x)
Simply, the x and y coordinates of a function are interchanged to get the inverse of the function.</span><span />
First, we can expand this power using the binomial theorem:
After that, we can apply De Moivre's theorem to expand each summand:
The final step is to find the common factor of i in the last expansion. Now:
The last part is to multiply these factors and extract the imaginary part. This computation gives:
(It is not necessary to do a lengthy computation: the summands of the imaginary part are the products sin(a)cos(b) and cos(a)sin(b) as they involve exactly one i factor)
A calculator simplifies the imaginary part Im(x⁶) to 0