Which one of the following options represents the given graph? A. f(x) = x * 2^(-x) B. f(x) = x * 2^(-|x|) C. f(x) = |x| * 2^(-x) D. f(x) = x * 2^(-x) (for x >= 0 only)

GATE 2023 · General Aptitude · Functions · medium

Answer: Option B: f(x) = x * 2^(-|x|)

1. Test each option for odd symmetry: A: f(-x) = (-x)*2^(x) != -f(x) = -x*2^(-x). Not odd. B: f(-x) = (-x)*2^(-|-x|) = -x*2^(-|x|) = -f(x). Odd. Passes. C: f(-x) = |-x|*2^(x) = |x|*2^(x) != -f(x). Not odd. D: Not defined for x<0 or not odd by structure.
2. Verify B decays in both directions: For x > 0: x * 2^(-x) -> 0 as x grows (exponential decay dominates). For x < 0: x * 2^(-|x|) = x * 2^x; as x -> -inf, 2^x -> 0 faster, so product -> 0. Both tails go to 0, confirming the graph shape.