Question

If $f(x+f(y\left)\right)=f\left(x\right)+y\forall x,y\in R$ and $f\left(0\right)=1$, then the value of $f\left(7\right)$ is

• A. $1$
• B. $7$
• C. $6$
• D. $8$

Answer 1

The correct option is A $1$

$f(x+f(y\left)\right)=f\left(x\right)+y,f\left(0\right)=1$
Putting $y=0$, we get
$f(x+f(0\left)\right)=f\left(x\right)+0$
or, $f(x+1)=f\left(x\right)\forall x\in R$
Thus, $f\left(x\right)$ is periodic with $1$ as one of its period.
Hence, $f\left(7\right)=f\left(6\right)=f\left(5\right)=\dots =f\left(1\right)=f\left(0\right)=1$