What is the amplitude of this function? What is the amplitude of the function f(x) = 2sin(x) + 2cos(x) ? 

Notice $$f(x)=2\sin x+2\cos x$$
$$=2(\sin x+\cos x)$$
$$=\frac{2\sqrt 2}{\sqrt 2}(\sin x+\cos x)$$
$$=2\sqrt{2}\left(\sin x\frac{1}{\sqrt 2}+\cos x \frac{1}{\sqrt 2}\right)$$
$$=2\sqrt{2}\left(\sin x\cos \frac{\pi}{4}+\cos x \sin\frac{\pi}{4}\right)$$
$$f(x)=2\sqrt{2}\sin \left(x+\frac{\pi}{4}\right)$$
The above function is of form $f(x)=x_0\sin (x+\phi)$ with amplitude $x_0$
hence, the amplitude of the given function $f(x)$ is $\color{red}{2\sqrt 2}$
