= Solution
In the two null coordinates the inverse metric has $g^{UV}=g^{VU}=-f^{-1}$ and $g^{UU}=g^{VV}=0$. Therefore
$$
\boxed{k^a=-(dU)^a=f^{-1}(\partial_V)^a,\qquad\ell^a=-(dV)^a=f^{-1}(\partial_U)^a.}
$$
Both tangents are null and future-directed, since $f>0$ and both coordinate directions are future-directed. To prove the <affine parameter> property, use the <null-gradient affine-geodesic identity>. The one-form $k_a=-\nabla_aU$ is closed, so $\nabla_ak_b=\nabla_bk_a$. Consequently
$$
k^a\nabla_ak_b=k^a\nabla_bk_a=\frac12\nabla_b(k^ak_a)=0.
$$
The same argument applies to $\ell_a=-\nabla_aV$. Thus both vector fields satisfy the <geodesic equation> without a nonaffinity term, and generate affinely parametrized <null geodesic congruences>. The coordinate vectors alone need not be affine; the factors $f^{-1}$ are essential.
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