Let $f:R\to R$ be a differentiable function that satisfies the relation $f(x+y)=f\left(x\right)+f\left(y\right)-1\forall x,y\in R,$ If $f^1\left(0\right)=2$  then $\left|f(-2)\right|$ is

$f(x+y)=f\left(x\right)+f\left(y\right)-1$

$f^1\left(x\right)=\underset{h\to 0}{lt}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lt}\frac{f\left(x\right)+f\left(h\right)-1-f\left(x\right)}{h}=\underset{h\to 0}{lt}\frac{f\left(h\right)-f\left(0\right)}{h}=f^1\left(0\right)=2$

$f^1\left(x\right)=2\Rightarrow \int dy=\int 2dx\Rightarrow y=2x+c$

If $x=0,y=1\Rightarrow c=1$

$\therefore y=2x+1$

$\left|f(-2)\right|=|-4+1|=3$