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Let f: R -> (0, pi) be defined as f(x) = cot⁻¹((2 - |x|)/(2 + |x|)). Which of the following statements are true?

1. f(x) is neither injective nor surjective
2. f(x) is continuous on R
3. f(x) is both an even function and an aperiodic function
4. lim_(x->inf) f(x) = lim_(x->-inf) f(x) = 3*pi/4

Correct answer: f(x) is continuous on R

Solution

Since f(x) depends only on |x|, it is an even function. f is continuous everywhere (cot⁻¹ is continuous and argument is continuous in |x|). The range: at x=0, f=pi/4; as |x|->inf, f->3pi/4. So f maps to [pi/4, 3pi/4) which is a subset of (0,pi) but not all of (0,pi), making it not surjective. Not injective either since f(-x)=f(x). The limit as x->+-inf equals 3pi/4. Statements B, C, D are all true; A is also true.

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