Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and $a=(a_1,\dots,a_n),\; a_n\neq 0$. Prove $a\bullet \nabla f(x) = 0 \;\forall x\in\mathbb{R}^n$ if and only if there's a differentiable $g:\mathbb{R}^{n-1}\to \mathbb{R}$ such that $$ f(x_1,\dots,x_n) = g(x_1-x_na_1/a_n,\dots,x_{n-1}a_{n-1}/a_n) $$
By assuming $f(x_1,\dots,x_n) = g(x_1-x_na_1/a_n,\dots,x_{n-1}a_{n-1}/a_n)$ and using chain rule, I already proved one direction. I'm stuck on how to construct such $g$ for the other direction.
Help please! Thank you!