I'm confused by Munkres' definition of the path product using the positive linear map. He defines the positive linear map $p: [a,b] \rightarrow [c,d]$ to be the unique map of the form $x \mapsto mx + k$ that carries $a$ to $c$ and $b$ to $d$. He says the following:
We can then describe the path product $f*g$ as follows: on $[0,\frac{1}{2}]$ it equals the positive linear map of $[0,\frac{1}{2}]$ to $[0,1]$ followed by $f$ and on $[\frac{1}{2},1]$ it equals the positive linear map of $[\frac{1}{2},1]$ to $[0,1]$ followed by $g$.
I feel the definition is unclear. What exactly does he mean "followed by $f$" and "followed by $g$"?