Let f(x)= {- a, if - a ≤ x ≤ 0; x+ a, if 0<x ≤ a} where a >0 and g (x)=(f(|x|)-|f(x)|) / 2. Then the function g:[- a, a ] →[- a, a ] is

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