Question

Let \(S \subseteq R\) and let \(f\) and \(g\) be functions from \(S\) into \(S\). Suppose that \(f\) is increasing on \(S\) and \(g\) is decreasing on \(S\). Is \(g \circ f\) necessarily increasing on \(S\), or decreasing on \(S\) or neither?

Answer 1

Definition 5.5.13 states that we are given a function \[f : X \rightarrow Y \] and a subset \[S \subseteq X\], the restriction of f to S is the function