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]