Functions \(f\) and \(g\) are defined, for \(x\gt 0\), by \(f(x)=\ln x\) and \(g(x)=2x^2+3\).

(i) Write down the range of \(f\).

(ii) Write down the range of \(g\).

(iii) Find the exact value of \(f^{-1}g(4)\).

(iv) Find \(g^{-1}(x)\) and state its domain.