Inverses of inverse functions
Inverses of inverse functions One more thing about inverse functions: if f has an inverse, it’s true that f−1(f(x)) = x for all x in the domain of f, and also that f(f−1(y)) = y for all y in the range of f. (Remember, the range of f is the same as the domain of f−1, so you can indeed take f−1(y) for y in the range of f without causing any screwups.) For example, if f(x) = x3, then f has an inverse given by f−1(x) = 3 √x, and so f−1(f(x)) = 3 √x3 = x for any x. Remember, the inverse function is like an undo button. We use x as an input to f, and then give the output to f−1; this undoes the transformation and gives us back x, the original number. Similarly, f(f−1(y)) = ( 3 √y)3 = y. So f−1 is the inverse function of f, and f is the inverse function of f−1. In other words, the inverse of the inverse is the original function. ...
