Determine the gamma distribution with parameters

Question

Solve the following problem:

The gamma distribution with parameters a and b, G(a, b), has density b^ax^a-1e^-bx/Γ(a). Show that if X ∼ IG(a, b), then 1/X ∼ G(a, b). (This means that generating from a gamma distribution is equivalent to generating from an inverted gamma distribution.

Make sure you use enough details to support your answer.