I'm having trouble understanding this question.
We have a path $h$ in $X$ from $x_0$ to $x_1$ and $\bar{h}$ its inverse path. Then a map $\beta_h:\pi_1(X,x_1)\to \pi_1(X,x_0)$ defined by $\beta_h[f]=[h\circ f\circ \bar{h}]$, for every path $f$ in $X$.
The question is to show that $\beta_h$ depends only on the homotopy class of $h$.
Firstly, it says for every path $f$ in $X$, but surely $f$ has to be a loop or you can't form $[h\circ f\circ \bar{h}]$?
And also, I don't understand why it depends on the homotopy class of $h$, when $[h\circ f\circ \bar{h}]$ is the path going from $x_0$ to $x_1$, around $f$, then back to $x_0$, why does the homotopy class of $h$ matter? In general I don't think I fully understand what this map $\beta_h$ is and would like someone to help me out. Thanks