Question

Let $f: R \rightarrow R$ be a differentiable function that satisfies the relation $f(x+y)=f(x)+f(y)-1, \forall x, y \in R$. If $f^{\prime}(0)=2$, then $|f(-2)|$ is equal to _____

Answer 1

$ f ( x + y )= f ( x )+ f ( y )-1 $
$ f^{\prime}(x)=\displaystyle\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $
$ f^{\prime}(x)=\displaystyle\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=f^{\prime}(0)=2$
$f^{\prime}(x)=2 \Rightarrow d y=2 d x $
$ y =2 x + C $
$ x =0, y =1, c =1 $
$ y =2 x +1 $
$ |f(-2)|=|-4+1|=|-3|=3$