Find all functions f(x): R R satisfying the equality x = -12f(|x|) + |f(x)| for all real numbers x.

f(x) = 2x for all x 0 and f(x) = 0 for all x < 0.

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69th Belarusian Mathematical Olympiad

Belarus algebra

Problem

Find all functions

f (x) : R \to R

satisfying the equality

x = - \frac{1}{2} f (∣ x ∣) + ∣ f (x) ∣

for all real numbers

x

Solution

f (x) = 2 x

for all

x \geq 0

and

f (x) = 0

for all

x < 0

Final answer

f(x) = 2x for x ≥ 0, and f(x) = 0 for x < 0

Techniques

Functional Equations