We have f(x) = 2x / 2(x^3 + x)

Let y = f(x) = 2x / 2(x^3 + x)

=> y = 2x / 2x(x^2 +1)

cancel 2x

=> y = 1/ x^2 + 1)

=> x^2 + 1 = 1/y

=> x^2 = 1/ y - 1

=> x^ 2 = (1 - y)/y

=> x = sqrt [ (1 - y)/y]

exchange x and y

=> y = sqrt [ (1 - x)/x]

The inverse function is

**f(x) = sqrt [ (1 - x)/x]**