Buseman function and isometry in Cheeger-Gromell splitting proof

Question

So I have the Busemann function $b^+$ as in the proof of the Cheeger-Gromell splitting theorem in Peterson and I want to show that if I have the isometry $f:(b^+)^{-1}(0)\times\mathbb{R}\rightarrow M$ defined by $f(p,t)=\exp_p(t\nabla b^+(p))$ then $b^+(f(p,t))=t$. Why is this assertion true?