Inverse of product of 2 or more functions

Question

I am interested to find the inverse of a product of 2 or more functions. For instance, suppose

$g(x) = \prod \limits_{i=1}^{n} [1 - x^{\alpha_i}], \quad x> 0, \alpha_i > 0$

Then, how do I find inverse of the function g? Is it the product of individual inverses?

Answer 1

There actually isn't a general formula for the inverse of $f\cdot g,$ where $f$ and $g$ are invertible functions. This is because the product of two invertible functions isn't necessarily invertible. For instance, on the nonzero real numbers, both $f(x) = x$ and $g(x) = \frac{1}{x}$ are invertible functions, but their product is $$f(x)\cdot g(x) = x\cdot \frac{1}{x} = 1,$$ which is neither injective nor surjective.

Answer 2

It appears that OP wants the inverse under multiplication (so, not under functional composition). So if $$g(x)=\prod_{i=1}^n(1-x)^{a_i}$$ then the inverse is given by $${1\over g(x)}={1\over\prod_{i=1}^n(1-x)^{a_i}}=\prod_{i=1}^n{1\over(1-x)^{a_i}}=\prod_{i=1}^n(1-x)^{-a_i}$$