2．The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sqrt { 1 - \mathrm { e } ^ { - x } } & x \in \mathbb { R } , x \geqslant 0
\mathrm {~g} ( x ) = \ln \left( 4 - x ^ { 2 } \right) & x \in \mathbb { R } , - 2 < x < 2 \end{array}$$ （a）（i）Explain why fg cannot be formed as a composite function．
（ii）Explain why gf can be formed as a composite function．
（b）（i）Find \(\mathrm { gf } ( x )\) ，giving the answer in the form \(\mathrm { gf } ( x ) = a + b x\) ，where \(a\) and \(b\) are constants．
（ii）State the domain and range of gf．
（c）Sketch the graph of the function gf．
On your sketch，you should show the coordinates of any points where the graph meets or crosses the coordinate axes． The circle \(C\) with centre \(( 0 , - \ln 9 )\) touches the line with equation \(y = \operatorname { gf } ( x )\) at precisely one point．
（d）Find an equation of the circle \(C\) ．