Find the range of f(x) = |sin x| + |cos x|, x ∈ R.

$f(x) = |\sin x| + |\cos x|\,∀\, x ∈ R$

Clearly, f(x) > 0.

Also, $f^2(x) = \sin^2 x + \cos^2 x + |2 \sin x \cos x| = 1 + |\sin 2x|$

Now, $0 ≤ |\sin zx|≤1$

$∴1 ≤ 1+|\sin zx|≤2$

$∴1 ≤ f^2(x)≤2$

or $1 ≤f(x)≤\sqrt{2}$