I am in highschool taking some advanced math courses and I have some questions about terminology.
There appears to be more definitions to the meaning of inversion in math than I can count. I'm hearing the word invert used in seemingly different ways without warning of switching the meaning.
I thought there was a single definition which was to take the reciprocal of a number. This definition seems to get almost no acknowledgement anymore.
Instead inversions are talked about in terms of inverting a function i.e. to solve for an input variable. Or I'm told to take the inverse of a number and add it to another when the meaning is to take the opposite of the number and add it e.g. the inverse of 3 --> -3.
It seems odd that the word inverse is being used where there are already words for said operation, like the latter example in the preceding paragraph. It also confuses me because I don't know whether to take the reciprocal or the opposite of the number and do something with it.
Lastly, the symbol inverse is used in notation for expressing the arctan, arcsin and arccos functions. Is there any intended correlation to the inverse definition in this use case?
A helpful answer here would include a list of all the different uses of the word inverse and when each of them are used. If possible, a definition of the word so that I can extrapolate its meaning if it's used in a context where I forget the precise use case.