Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0< x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}> 0$ and $\mathrm{g}(x)=(f(|x|)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is