Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by\n\(\nf:x\mapsto x-a,\quad x\in\mathbb{R},\quad -a\leq x\leq 2a,\n\)\n\(\ng:x\mapsto \ln x,\quad x\in\mathbb{R},\quad x>0,\n\)\nwhere \(a\) is a positive constant. Explain why the composite function \(\mathrm{gf}\) does not exist.[2]
The function \(\mathrm{h}\) is defined by\n\(\nh:x\mapsto |(x-b)(b+2-x)|,\ x\in\mathbb{R}\n\)\nwhere \(b\) is a positive constant.
Sketch the graph of \(y=\mathrm{h}(x)\), giving in terms of \(b\), the values of any points where the curve meets the \(x\)-axis.[2]
The domain of \(\mathrm{h}\) is further restricted to \(x\geq k\), where \(k\) is a constant.\n\nState the least value of \(k\) in terms of \(b\) such that the function \(\mathrm{h}^{-1}\) exists.[1]
Hence find \(\mathrm{h}^{-1}(x)\) in terms of \(b\).[3]
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