Discussion
Which one of the following options represents the given graph?
Answer: A
The graph is \(f(x) = x^{2}\,2^{-|x|}\).
Read three features from the graph: it is symmetric about the vertical axis (an even function), it is never negative and touches zero at \(x = 0\), and it decays to zero on both sides.
\(x^{2}\,2^{-|x|}\) has all three: \(x^{2}\) and \(|x|\) are both even, the product is non-negative and zero at the origin, and \(2^{-|x|}\) makes it vanish as \(|x|\) grows, with peaks in between.
\(x\,2^{-|x|}\) is odd, so it would be negative for \(x \lt 0\). \(|x|\,2^{-x}\) and \(x\,2^{-x}\) contain \(2^{-x}\), which grows without bound as x becomes large and negative, so they cannot fall back to zero on the left. Only the first option fits.
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