Matching rules are not equal without matching domains

The function $f$ is defined by $f:x\mapsto e^{|2-x|}$, $x\in\mathbb R$, $x>a$, where $a$ is a constant. It is given that $a$ is the least possible value such that $f^{-1}$ exists.

(i) State the value of $a$.1 mark

(ii) Hence find $f^{-1}(x)$ and state the domain of $f^{-1}$.4 marks

(iii) Sketch on the same diagram the graphs of $y=f(x)$, $y=f^{-1}(x)$ and $y=ff^{-1}(x)$, giving the coordinates of all end points.3 marks

It is now given that $a$ is replaced by a constant such that the functions $ff^{-1}$ and $f^{-1}f$ are equal.

(iv) Explain why the new value of $a$ satisfies the equation $a=f(a)$, and find $a$.2 marks

(v) The function $g$ is defined by $g:x\mapsto e^{|2-x|}$, $x\in\mathbb Z$. Explain why composite $g^2$ does not exist.1 mark

[2021/EJC/Promo/8]