The functional inequality $f(|x|)+f(|y|) \geq 1/f(|x+y|)$

Question

Please help me to solve the following problem.

Does there exist a nonempty function $f: D_{f} \subset \mathbb{R} \to \mathbb{R}$ with $D_{f} \neq \emptyset$ such that $$ f(|x|)+f(|y|) \geq \frac{1}{f(|x+y|)} $$ for all $x, y \in \mathbb{R}$ such that $|x|, |y|, |x+y| \in D_{f}$ ?

Thank you very much in advance!

Answer 1

What about $D_f = \Bbb{R}$ and $$ \forall x \in \Bbb{R} : f(x) = 1 $$

Am I missing something, because this seems too simple.