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

References

1. A dictionary of science, literature and art, ed. by W.T. Brande. Pg 683
2. A dictionary of science, literature and art, ed. by W.T. Brande. Pg 683
3. A dictionary of science, literature and art, ed. by W.T. Brande. Pg 683

• Mathematics