The diagram shows the graph of \(y=\mathrm{f}(x)\) with asymptotes \(x=1\) and \(y=2\), and a minimum point at \(A\left(-2,-\dfrac{3}{2}\right)\). The curve cuts the \(x\)-axis at \((0,0)\) and \((-3,0)\). On separate diagrams, sketch the graphs of
\(y=\dfrac{1}{\mathrm{f}(x)}\),[3]
\(y=\mathrm{f}'(x)\),[3]
stating, in each case, the equations of asymptotes, the exact coordinates of any axial intercepts, turning points and points corresponding to \(A\) where possible.
The function \(\mathrm{g}\) is defined by\n\(\ng(x)=\begin{cases}2\cos\left(\dfrac{\pi x}{4}\right)+2&\text{ for}\hspace{0.5em}0\le x<4,\\x-2&\text{ for}\hspace{0.5em}4\le x<6,\end{cases}\n\)\nand \(\mathrm{g}(x)=\mathrm{g}(x+6)\) for \(x\in\mathbb{R}\).\n\nSketch the graph of \(y=\mathrm{g}(x)\) for \(-6\le x<4\), showing clearly the end points values. Find the exact value of \(\mathrm{g}(2025)\).[4]
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