Determine the range of f(x) = 2 extcos x + 1 and the period of g(x) = extsin 2x.

The function is $f(x) = 2\cos x + 1$.
The range of the cosine function is $-1 \le \cos x \le 1$.
Multiply by 2: $-2 \le 2\cos x \le 2$.
Add 1: $-2+1 \le 2\cos x + 1 \le 2+1$.
So, $-1 \le f(x) \le 3$.

The range of $f$ is $\boxed{[-1, 3]}$.

The function is $g(x) = \sin 2x$.
For a function of the form $y = \sin(bx)$, the period is $\frac{360^\circ}{|b|}$.
Here, $b=2$.

$$\text{Period} = \frac{360^\circ}{2} = 180^\circ$$

The period of $g$ is $\boxed{180^\circ}$.