Invertible Functions are Exponentials of compactly supported functions?

Question

Let $C'(\mathbb{R}^d)$ be the set of all continuous functions from $\mathbb{R}^d$ to itself. If $g \in C'(\mathbb{R}^d)$ has the property that $g(x)=x$ outside the closed unit ball and $g$ fixed the origin , then can we necessarily represent $g$ as $$ g = exp( Af(|x|) +b ) x, $$ where $A$ is a $d\times d$ invertible matrix $b\in \mathbb{R}^d$ and $f$ is acontinuous and compactly supported function, and $\exp$ is the matrix exponential.