91Ó°ÊÓ

We are given two increasing functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\), meaning that if \(x < y\), then \(f(x) < f(y)\) and \(g(x) < g(y)\) hold for any \(x, y \in \mathbb{R}\). We want to show that the composition \((g \circ f): \mathbb{R} \rightarrow \mathbb{R}\), defined as \((g \circ f)(x) = g(f(x))\), is also an increasing function. Since \(f(x) < f(y)\), we can apply the increasing function \(g\) on both sides to get \(g(f(x)) < g(f(y))\). Rewriting the inequality in terms of the composition, we get \((g \circ f)(x) < (g \circ f)(y)\) for any \(x, y \in \mathbb{R}\) such that \(x < y\), concluding that the composition of two increasing functions is indeed an increasing function.