GATE 2023 XH5 (Psychology) – Question 7

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

Which one of the following options represents the given graph?

A curve through the origin. For positive $x$ it rises quickly to a maximum close to the origin and then decays slowly towards zero. For negative $x$ it is the mirror image through the origin: it falls to a minimum and then returns to zero.

$f(x) = x^2\, 2^{-|x|}$
$f(x) = x\, 2^{-|x|}$
$f(x) = |x|\, 2^{-x}$
$f(x) = x\, 2^{-x}$

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Correct answer: (B) $f(x) = x\, 2^{-|x|}$

Explanation

The curve is an odd function: $f(-x) = -f(x)$, and it tends to 0 for large $|x|$. $x\,2^{-|x|}$ satisfies both. $x^2 2^{-|x|}$ and $|x| 2^{-x}$ are not odd (the first is even and positive), and $x\,2^{-x}$ grows without bound for negative $x$.