A non-invertible function can have an invertible composite

$f:x\mapsto(x-7)(x+1)$, $x>0$.

(a) Use an algebraic method to find exact solution of $f(x)<(x^2-7x)/(2-x)$.5 marks

(b)(i) Determine if $f^{-1}$ exists.2 marks

Given $g$ is one-one with domain $(0, \infty)$ and range $(4, \infty)$:

(ii) Find range of $fg$.2 marks

(iii) A function $h$ is one-one if for any $\alpha, \beta\in D_h$, $h(\alpha)=h(\beta)$ implies $\alpha=\beta$. Show that $fg$ is one-one.2 marks

(iv) If $g(x)=4e^x$ for $x>0$, find exact $(fg)^{-1}(20)$.4 marks

[2011/RI/Promo/13]