real analysis - Does there exist a scalar function $g({bf{x}})$ that satisfies $g({bf{x}} +,{bf{f}}({bf{x}}))= g ({bf{x}})det(I+,{bf{f}}'({bf{x}}))$?

Given a vector valued function $\bf{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$, what is a scalar function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies the following $$g({\bf x} +\mathbf{f}(\mathbf{x}))=g(\mathbf{x})\det(I+\mathbf{f}'(\mathbf{x})),$$ where $\bf{f}'(\bf{x})$ is the Jacobian matrix of $\bf{f}(\bf{x})$ and $I$ is the $n\times n$ identity matrix.

If a solution can't be found for arbitrary $\bf{f}$, what structure can one impose on $\bf{f}$ for there to exist a particular function $g$ that satisfies the above condition.

One particular example of a function $g$ that satisfies the above is all I'm after (i.e., I don't need the most general solution).

Alternatively: Can one prove that there does not exist a function $g$ that satisfies the above?