Problem

The diagram shows the graph of $y = f(x)$ where the function $f$ is defined by

$$f(x) = 3 + 2\sin\frac14 x \quad \text{for } 0 \leq x \leq 2\pi.$$

(a) On the diagram above, sketch the graph of $y = f^{-1}(x)$.

(b) Find an expression for $f^{-1}(x)$.

(c)

The diagram above shows part of the graph of the function $g(x) = 3 + 2\sin\frac14 x$ for $-2\pi \leq x \leq 2\pi$.

Complete the sketch of the graph of $g(x)$ on the diagram above and hence explain whether the function $g$ has an inverse.

(d) Describe fully a sequence of three transformations which can be combined to transform the graph of $y = \sin x$ for $0 \leq x \leq \frac12\pi$ to the graph of $y = f(x)$, making clear the order in which the transformations are applied.