The function \(\mathrm{f}\) is defined by \[\mathrm{f}:x\mapsto\frac{x-1}{x+1},\qquad x\in\mathbb{R}\setminus\{-1,0,1\}.\]
State the range of \(\mathrm{f}\) and find the rule of \(\mathrm{f}^2\).[3]
Hence, or otherwise, show that \(\mathrm{f}^4(x)=x\).[2]
Find the exact value of \(\mathrm{f}^{2027}(5)\).[2]
The function \(\mathrm{g}\) is defined by \[\mathrm{g}:x\mapsto x^2+a,\qquad x\in\mathbb{R},\quad a\in\mathbb{R}^+.\]
Find the range of \(\mathrm{gf}\) in terms of \(a\).[2]