= Solution
For radial <null geodesic>[null geodesics], $d\Omega_3=dz=0$ and $ds^2=0$, so
$$
\frac{dt}{dr}=\pm\frac1{f(r)}.
$$
Introduce the tortoise coordinate by
$$
\frac{dr_\star}{dr}=\frac1f,
\qquad
\boxed{r_\star=r+\frac{r_+}{2}\log\left|\frac{r-r_+}{r+r_+}\right|}
$$
up to an additive constant. The plus sign gives $d(t-r_\star)=0$ on outgoing rays, while the minus sign gives $d(t+r_\star)=0$ on ingoing rays. Hence
$$
\boxed{u=t-r_\star},
\qquad
\boxed{v=t+r_\star}
$$
are respectively constant on the stated radial null families.
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