AMSOC 26-003 Module A Practice Test

When a function is strictly increasing and invertible, its inverse must preserve the order of values. If you take two values in the range of f, say y1 < y2, these come from some x1 and x2 with y1 = f(x1) and y2 = f(x2). Because f is strictly increasing, x1 < x2. But x1 = f^{-1}(y1) and x2 = f^{-1}(y2), so f^{-1}(y1) < f^{-1}(y2). This shows f^{-1} is strictly increasing on its domain, which is the range of f. The inverse cannot be decreasing, not monotonic, or constant, given the one-to-one, order-preserving nature of f.