Mathematics

Mathematics

Problem

Let f:R→Rf:\mathbb R\to\mathbb R and g:R→Rg:\mathbb R\to\mathbb R be bijective functions satisfying g∘f=f∘gg\circ f=f\circ g. a. (i) Prove that f−1∘g=g∘f−1f^{-1}\circ g=g\circ f^{-1}. b. (ii) Hence show that (g∘f)−1=f−1∘g−1=g−1∘f−1(g\circ f)^{-1}=f^{-1}\circ g^{-1}=g^{-1}\circ f^{-1}.
Intuitive explanation

The key insight is that ff and gg commute, meaning they can be applied in either order with the same result. This severely constrains their structure—you're looking for what kinds of functions can "play nicely" together this way. Think about how linear functions or power functions with the same base behave: they often commute because they share an underlying symmetry or operate on the same "scale." The challenge is to characterize all bijections with this property, which often means finding a common structure they must have (like both being powers of a third function, or both being affine transformations of a specific form).

Intuitive explanation

The key insight is that ff and gg commute, meaning they can be applied in either order with the same result. This severely constrains their structure—you're looking for what kinds of functions can "play nicely" together this way. Think about how linear functions or power functions with the same base behave: they often commute because they share an underlying symmetry or operate on the same "scale." The challenge is to characterize all bijections with this property, which often means finding a common structure they must have (like both being powers of a third function, or both being affine transformations of a specific form).

Browse the problem library