Explicit path homotopy from a path followed by its reverse to the constant path

Let \(f:I\to X\) be a path in a topological space \(X\), and define the reverse path by \[ \overline f(s)=f(1-s). \] Let \(e_{x_0}\) be the constant path at \(x_0=f(0)\). Construct an explicit path homotopy, described algebraically and in full detail, proving \[ [f]*[\overline f]=[e_{x_0}]. \]