The function \(\mathrm h\) is defined by \(\mathrm h(x)=\sqrt{16-(x-3)^2}\), \(x\in(a,b)\), where \(a,b\in\mathbb R\).
Find the value of \(a\) and of \(b\) such that the domain \(D_{\mathrm h}=(a,b)\) is the largest possible.[2]
Find the equation of the line of symmetry on the graph \(y=\mathrm h(x)\).[2]
The graph of \(\mathrm h\) goes through a translation of \(\begin{pmatrix}p\\0\end{pmatrix}\) such that the resulting graph is an even function.
State the value of \(p\).
Show that \(\mathrm h(x-p)\) is an even function.[4]
Given another function \(\mathrm g(x)=\dfrac1x\), \(x\in\mathbb R\), \(x\neq0\), find
the rule for the function \(\mathrm g\circ\mathrm h\),
the range of the function \(\mathrm g\circ\mathrm h\).[4]
Let \(\mathrm k(x)=-\mathrm h(3x)\).
The point \(A(m,2)\) on the graph of \(\mathrm h\) is mapped to the point \(A'\) on the graph of \(\mathrm k\).
Find the possible values of \(m\).
Hence find the corresponding coordinates of \(A'\).[4]