find \({f^{ - 1}}(x)\), stating its domain;

find the value of x such that \(f(x) = {f^{ - 1}}(x)\).

\(y = \frac{1}{{1 + {{\text{e}}^{ - x}}}}\)

\(y(1 + {{\text{e}}^{ - x}}) = 1\)     M1

\(1 + {{\text{e}}^{ - x}} = \frac{1}{y} \Rightarrow {{\text{e}}^{ - x}} = \frac{1}{y} - 1\)     A1

\( \Rightarrow x = - \ln \left( {\frac{1}{y} - 1} \right)\)     A1

\({f^{ - 1}}(x) = - \ln \left( {\frac{1}{x} - 1} \right)\,\,\,\,\,\left( { = \ln \left( {\frac{x}{{1 - x}}} \right)} \right)\)     A1

domain: 0 < x < 1     A1A1

Note: Award A1 for endpoints and A1 for strict inequalities.

[6 marks]

0.659     A1

[1 mark]

Finding the inverse function was done successfully by a very large number of candidates. The domain, however, was not always correct. Some candidates failed to use the GDC correctly to find (b), while other candidates had unsuccessful attempts at an analytic solution.

Finding the inverse function was done successfully by a very large number of candidates. The domain, however, was not always correct. Some candidates failed to use the GDC correctly to find (b), while other candidates had unsuccessful attempts at an analytic solution.