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What is the preferred notation for expressing the first $x$ where $f(x)$ is greater than a threshold $t$. This is similar to $\arg\max$ notation but instead of max, I want the first $x$ where $f(x)$ is greater than $t$.

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I don't think there's a standard notation for this concept specifically, but one could express it as $\inf f^{-1}((t,\infty)).$ Alternately, as Henning points out below, we could write $\min\{x\mid f(x)>t\},$ which is indeed significantly clearer.

Note that using $\inf$ instead of $\min$ guarantees the quantity exists, but we may not actually have $f(a)>t$ where $a=\inf\{x\mid f(x)>t\}$.

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    These are not so much *notations* as formulas for what's to be denoted.2011-12-02
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My prefered notation for the first $x$ where $f(x)$ greater than $t$ is “the first x where f(x) greater than t”.