Need help understand how to sketch the graph of reciprocals.

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sketch of f(x)

sketch of 1/f(x)

First picture is $y=f(x)$, second picture is $y=\frac{1}{f(x)}$

For this type of question, i have never understood how to determine the concavity of the reciprocal function? From $0<x<a$, there is a point of inflexion, but how does the original qraph gives us this. If anything, the $\cap$-looking curve in the original graph would suggest to me that the reciprocal should also be symmetrical between $a$ and $-a$ but yet they clearly aren't.