The function \(f\) is defined by

\(f(x)=\frac{\sqrt{4x^2-1}}{2x}\quad\text{for }0.5\leqslant x\leqslant1.5.\)

The diagram shows a sketch of \(y=f(x)\).

(a)

(i) It is given that \(f^{-1}\) exists. Find the domain and range of \(f^{-1}\).

(ii) Find an expression for \(f^{-1}(x)\).

(b) The function \(g\) is defined by \(g(x)=e^{x^2}\) for all real \(x\). Show that \(gf(x)=e^{\left(1-\frac{a}{bx^2}\right)}\), where \(a\) and \(b\) are integers.