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.
                            Now, you have to be careful in the case where you restrict the domain. Let g(x) = x2; we’ve seen that you need to restrict the domain to get an inverse. Let’s say we restrict the domain to [0,∞) and carelessly continue to refer to the function as g instead of h, as in the previous section. We would then say that g−1(x) = √x. If you calculate g(g−1(x)), you find that this is (√x)2, which equals x, provided that x ≥ 0. (Otherwise you can’t take the square root in the first place.)
                             On the other hand, if you work out g−1(g(x)), you get √x2, which is not always the same thing as x. For example, if x = −2, then x2 = 4 and so √x2 = √4 = 2. So it’s not true in general that g−1(g(x)) = x. The problem is that −2 isn’t in the restricted-domain version of g. Technically, you can’t even compute g(−2), since −2 is no longer in the domain of g. We really should be working with h, not g, so that we remember to be more careful. Nevertheless, in practice, mathematicians will often restrict the domain with- out changing letters! So it will be useful to summarize the situation as follows:
If the domain of a function f can be restricted so that f has an inverse f−1, then
• f(f−1(y)) = y for all y in the range of f; but
• f−1(f(x)) may not equal x; in fact, f−1(f(x)) = x only when x is in the restricted domain.