Functions \(f\) and \(q\) are defined by
\(f : x \mapsto e ^{(x-1)^{2}}, \quad x \in R\)
\(g : x \mapsto \frac{1}{2-x}, \quad x \in R , \quad 1 \leq x<2\)
(i) Sketch the graph of \(y= f (x)\).
[1]
(ii) If the domain of \(f\) is restricted to \(x \geq k\), state with a reason the least value of \(k\) for which the function \(f^{-1}\) exists.
[2]
In the rest of the question, the domain of \(f\) is \(x \geq k\), using the value of \(k\) found in part (ii).
(iii) Find \(g ^{-1}(x)\) and show that the composite function \(g ^{-1} f ^{-1}\) exists.
[4]
(iv) Find the range of \(g ^{-1} f ^{-1}\).
[1]
(i)

(ii) For \(f ^{-1}\) to exist, \(f\) must be a one-one function. Least value of \(k=1\).
(iii) \(g ^{-1}(x)=2-\frac{1}{x}\)
(iv) \(R_{g^{-1} f^{-1}}=[1,2)\)
