Problem

Functions $f$ and $g$ are defined as follows.

$f(x)=\cos x\quad\text{for }0\leq x\leq\pi$

$g(x)=3\cos(x-\pi)+2\quad\text{for }\pi\leq x\leq2\pi$

(a) Describe fully the transformations that have been combined to transform the graph of $y=f(x)$ to the graph of $y=g(x)$.

(b) On the given axes, sketch the graphs of $y=f(x)$ and $y=g(x)$.

(c) Find $g^{-1}f\left(\frac{1}{3}\pi\right)$.

(d) Explain why the composite function $fg$ cannot be formed.