 Solution

The normal to $r=r_{+}$ is $dr$, whose squared norm is $g^{rr}=f(r_{+})=0$, so the surface is a null hypersurface. In ingoing coordinates $K={\partial }_v$, and on the horizonThus $K$ is both tangent and normal there, making the surface a Killing horizon. For a static metric of this form, the surface gravity is $\kappa =f^{\prime }(r_{+})/2$. With $n=d-3$,