Which one of the following options represents the given graph? [Graph shows a smooth bell-shaped curve with a single peak above the x-axis, symmetric about x = 0, approaching zero as x -> +/-inf] A. f(x) = x * 2^(-x^2) B. f(x) = x * 2^(|x|) C. f(x) = |x| * 2^(-x^2) D. f(x) = x * 2^(-x)

GATE 2023 · General Aptitude · Functions · medium

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

1. Check the graph properties: The graph shows: f(0) = 0, f(x) > 0 for x in small positive range, f(x) -> 0 as x -> +inf, and the function is negative (or has a trough) for x < 0. This is an odd-like function with Gaussian-type decay.
2. Analyze f(x) = x * 2^(-x^2): f is an odd function. f(0) = 0. For x > 0, f(x) > 0 with a single maximum. For x < 0, f(x) < 0 with a single minimum. As x -> +/-inf, 2^(-x^2) -> 0 faster than x grows, so f -> 0.
3. Eliminate other options: B: x * 2^|x| grows without bound. C: |x| * 2^(-x^2) is even (non-negative, two peaks). D: x * 2^(-x) is not symmetric and does not decay for x < 0.