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\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-10_784_913_274_607}
The diagram shows the graph of \(y = \mathrm { f } ( x )\) where the function f is defined by
$$f ( x ) = 3 + 2 \sin \frac { 1 } { 4 } x \text { for } 0 \leqslant x \leqslant 2 \pi$$
- On the diagram above, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
- Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-11_759_1545_276_331}
The diagram above shows part of the graph of the function \(\mathrm { g } ( x ) = 3 + 2 \sin \frac { 1 } { 4 } x\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).
Complete the sketch of the graph of \(\mathrm { g } ( x )\) on the diagram above and hence explain whether the function \(g\) has an inverse.- Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) to the graph of \(y = \mathrm { f } ( x )\), making clear the order in which the transformations are applied.