The number of linear function $$f$$ satisfying $$f\left( {x + f\left( x \right)} \right) = x + f\left( x \right)\,\forall \,x\, \in \,R$$       is :

Solution :
$$\eqalign{ & {\text{Let }}f\left( x \right) = ax + b.....(1) \cr & \Rightarrow f\left( {ax + b + x} \right) = x + ax + b \cr & \Rightarrow f\left( {\left( {a + 1} \right)x + b} \right) = \left( {a + 1} \right)x + b \cr & {\text{Replace }}\left( {a + 1} \right)x + b\,{\text{by }}y{\text{, we have}} \cr & \Rightarrow f\left( y \right) = \left( {a + 1} \right)\left( {\frac{{y - b}}{{a + 1}}} \right) + b \cr & {\text{or }}f\left( x \right) = \left( {a + 1} \right)\left( {\frac{{x - b}}{{a + 1}}} \right) + b \cr} $$
$$\therefore $$  required number of linear functions is 2.