Composition of 2 monotonic functions

Question

Let $f$ be a monotonic function $f:[a,b] \rightarrow\mathbb{R}$ and $g$ be a monotonic function $g:[c,d]\rightarrow[a,b]$. Show that $f\circ g$ is monotonic

Answer 1

Let $*$ represent either $<$ or $>$, depending on the direction of the monotonicity in each case. Then $$x*y\implies g(x)*g(y)\implies f\circ g(x)*f\circ g(y)$$