Functional notation

In functional notation, a letter, as a symbol of operation, is combined with another which is regarded as a symbol of quantity. Thus $f(x)$ denotes the mathematical result of the performance of the operation $f$ upon the subject $x$. If upon this result the same operation were repeated, the new result would be expressed by $f[f(x)]$, or more concisely by $f^{2}(x)$, and so on. The quantity $x$ itself regarded as the result of the same operation $f$ upon some other function; the proper symbol for which is, by analogy, $f^{-1}(x)$. Thus $f$ and $f^{-1}$ are symbols of inverse operations, the former cancelling the effect of the latter on the subject $x$. $f(x)$ and $f^{-1}(x)$ in a similar manner are termed inverse functions.