Consider the function f , where \(f(x) = \arcsin (\ln x)\).

(a)     Find the domain of f .

(b)     Find \({f^{ - 1}}(x)\).

(a)     \( - 1 \leqslant \ln x \leqslant 1\)     (M1)

\( \Rightarrow \frac{1}{{\text{e}}} \leqslant x \leqslant {\text{e}}\)     A1A1

(b)     \(y = \arcsin (\ln x) \Rightarrow \ln x = \sin y\)     (M1)

\(\ln y = \sin x \Rightarrow y = {{\text{e}}^{\sin x}}\)     (M1)

\( \Rightarrow {f^{ - 1}}(x) = {{\text{e}}^{\sin x}}\)     A1

[6 marks]

Very few candidates attempted part (a), and of those that did, few were successful. Part (b) was answered fairly well by most candidates.