Marc D. answered • 11/20/20
Engaging and patient Master of Applied Mathematics.
You can define the inverse function as g(x) = inverse(f(x)).
Since g(x) is a function, so is the inverse of f(x).
I think for most introductory courses, this definition would suffice, but if you would like a little more information, read on!
There are nuances such as whether the function is one-to-one or "injective" which means that for every element in the domain of the function maps to a specific element in the co-domain.
or, if the function is surjective, which means for every element in the target set, there is at least one element in the domain.
This might seem tedious, but if your lecturer has defined a function as something that maps a unique element from the domain to a unique element in the range, and that all elements in the range are mapped to, then the function would be called bijective, having both surjection and injection. in that case, the inverse function would also match the definition.
If the given function is surjective, and he expects that the definition of a function is bijective, then the inverse would not be a function, since the inverse function would map a single element from the co-domain back to more than 1 element in the domain of the original function.
I hope that makes things a little clearer. Check out : https://en.wikipedia.org/wiki/Injective_function for some diagrams.
