Find the components f(x) and g(x) so that fog(x)=H(x) where H(x)=ln(x^3-8)

The function H(x) = ln(x^3 - 8). We have to determine two functions f(x) and g(x) such that fog(x) = H(x).

There can be many options for f(x) and g(x) such that fog(x) = H(x), though the simplest is: f(x) = ln (x) and g(x) = x^3 - 8

fog(x) = f(g(x)) = f(x^3 - 8) = ln(x^3 - 8) which is the same as H(x).

Also, for g(x) the domain is R and that of f(x) is (2, inf.) which makes the domain of f(g(x)) all values of x such that x > 2. The domain of H(x) is also (x, inf.).

The required functions are f(x) = ln x and g(x) = x^3 - 8