Proving homotopy of paths

Question

Let $f$ be a path in $X$ and $h:[0,1] \mapsto [0,1]$ a continuous mapping with $h(0)=0$ and $h(1)=1$. How can I prove that $f$ and $fh$ are homotopic relative to the endpoints?

Answer 1

Consider the homotopy H(s,t)=f(ts+(1-t)h(s)). Hope it helps

Answer 2

Hint: Clearly there is a homotopy $H$ between $h$ and the identity function $x$ on $[0,1]$. Now consider $f\circ H$.