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I am encountering functions of real variable with the following property: $$ f(x) = f(1/x) $$ For example, $$ f(x) = \left(x - \frac{1}{x}\right)\log^{3}{x} \qquad x > 0 $$ Is there a name for this property?

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    "Symmetric with respect to reciprocation" is how I would call it2011-10-04
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    Invariant under inversion2011-10-04
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    This is an observation, not a name. If the domain of $f$ consists of positive numbers, then $g(t) = f(e^t)$ is symmetric about the origin.2011-10-04
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    @SrivatsanNarayanan: I think you mean $f(e^t)$ is symmetric across the $y$ axis.2011-10-04
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    @Shaun Ah, yes. You are correct. I thought of it as a function, rather than as a graph on 2-D. In any case, it might be best to say "$g(t)$ is an even function".2011-10-04

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Summary of comments:

  • "Invariant under inversion" is a good name for such functions
  • If the domain consists of positive numbers, one can say instead "$t\mapsto f(e^t)$ is even".