= Solution
Reversing sign reverses the inequality. Since $X$ has a <probability density function>, it has no point masses, so
$$
F_Y(y)=\mathbb P(X\geq-y)=1-\mathbb P(X<-y)=\boxed{1-F_X(-y)}.
$$
Differentiating the <cumulative distribution function> gives
$$
\boxed{f_Y(y)=f_X(-y)}
$$
almost everywhere: the minus sign outside $F_X$ cancels the minus sign from the <chain rule>. If point masses were allowed, the <cumulative distribution function> would instead require the left limit $1-F_X((-y)^-)$.
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