10. Given that $f(x)$, $g(x)$, and $h(x)$ are all linear functions. If for real numbers $x$ the following holds:
$$
\begin{array}{l}
|f(x)|-|g(x)|+h(x) \\
=\left\{\begin{array}{ll}
-1, & x<-1 ; \\
3 x+2, & -1 \leqslant x<0 ; \\
-2 x+2, & x \geqslant 0,
\end{array}\right.
\end{array}
$$

then $h(x)=(\quad)$.
(A) $x-\frac{1}{2}$
(B) $-x-\frac{1}{2}$
(C) $-x+\frac{1}{2}$
(D) $x+\frac{1}{2}$
10. C.
$$
h(x)=\frac{-2 x+2+(-1)}{2}=-x+\frac{1}{2} .
$$