Given \[ f_{x}\left\{\begin{array}{ll} x & x<-2 \\ x+1 & -2 \leq x<2 \\ x^{2}-2 & x \geq 2 \end{array}\right. \] find $f(0)$

Step 1 ：Given the function: \[f_{x}\left\{\begin{array}{ll} x & x<-2 \\ x+1 & -2 \leq x<2 \\ x^{2}-2 & x \geq 2 \end{array}\right.\]

Step 2 ：We need to find the value of $f(0)$.

Step 3 ：The value $0$ falls into the range $-2 \leq x < 2$. Therefore, we need to substitute $0$ into the expression $x+1$ to find $f(0)$.

Step 4 ：Substituting $x = 0$ into the expression $x+1$, we get $f_x = 1$.

Step 5 ：Final Answer: $f(0) = \boxed{1}$