RI/2021/JC2/Prelim/P2/Q01

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]