Restrict a parabola and test an inverse composite

The function $f$ is defined by $f:x\mapsto5-(x-2)^2$, $x\leq4$.

(a) Sketch the graph of $y=f(x)$. Hence state the range of $f$.2 marks

(b) Explain why $f^{-1}$ does not exist.1 mark

The function $g$ is such that $g:x\mapsto f(x)$, $x\leq k$, where $k$ is a real constant.

(c) Given that $g^{-1}$ exists, state the largest possible value of $k$.1 mark

Use the largest possible value of $k$ found in part (c) to answer the remaining parts of this question.

(d) Find $g^{-1}$ in similar form.3 marks

(e) The function $h$ is defined by $h:x\mapsto\dfrac{\alpha x+1}{x-2}$, $x\in\mathbb R$, $x\ne2$ where $\alpha$ is a positive constant.

(i) By finding the range of $h$ in terms of $\alpha$, determine the range of values of $\alpha$ such that $h^{-1}g$ exist.3 marks

(ii) Solve $h^{-1}g(x)=-11$ for the case where $\alpha=6$.2 marks