As a permutation is a bijection, then it has an inverse which is also a
bijection. The inverse merely undoes what the original function does.
Thus if 
then its inverse 
maps j to i.
To find the inverse of a permutation that is a cycle all we have to do is write the elements of the cycle in reverse order. Thus the inverse of (1 2 3 4) is (4 3 2 1). Since a cycle can be written with any of its elements as the first term we can also write this inverse as (1 4 3 2). This gives an alternative way to write down the inverse of a cycle. Fix the first element in the cycle and write the remaining elements in reverse order. Thus, the inverse of (1 2 3 4 5) is (1 5 4 3 2).
In the case that the permutation is a product of cycles we must reverse the order of the cycles as well as invert each cycle. As an example, the inverse of (1 3 6 2)(3 7)(2 4 5) is (5 4 2)(7 3)(2 6 3 1) which can also be wriiten as (2 5 4)(3 7)(1 2 6 3).
For permutations written as the product of disjoint cycles we need only invert each cycle. For example, the inverse of (1 2 3)(5 6)(9 10) is (3 2 1)(6 5)(9 10) which can also be written as (1 3 2)(5 6)(9 10). Note that cycles of length two do not have to be inverted.
One might ask why we don't write the cycles in reverse order when finding the inverse of a permutation written as a product of disjoint cycles. The answer is that you can do that but why? Since the cycles are disjoint then they all commute with each other and so the order of the disjoint cycles does not matter. For example, the inverse of (1 5 2 3)(4 6 7) is (7 6 4)(3 2 5 1). (Note this reverses the elements in each cycle and the order of the cycles.) Now as the cycles are disjoint the inverse can be written as (3 2 5 1)(7 6 4) and the effect is the same as just inverting each cycle.
Note that if we want to find the inverse of a permutation consisting of non disjoint cycles we could first multiply out the cycles and then proceed to find the inverse of the permutation as a product of disjoint cycles. For example we found the inverse of (1 3 6 2)(3 7)(2 4 5) to be (5 4 2)(7 3)(2 6 3 1). Now we'll find the inverse in a different way. First multiply out the cycles in (1 3 6 2)(3 7)(2 4 5) to get (1 7 3 6 4 5 2) and now invert to get (1 2 5 4 6 3 7). While this does look different to the inverse we found it is actually identical and we can see that by writing the inverse (5 4 2)(7 3)(2 6 3 1) in disjoint cycle form. That is done by multiplying out the cycles and we get (1 2 5 4 6 3 7) as before.
One can always check if the inverse is correct by multiplying the two permutations to see if their product is 1