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.