Let f be a non-zero real valued continuous function satisfying f(x+y)=f(x) f(y) for all x, y R . If f(2)=9 , then f(6) is equal to

Step-by-step solution

f(x+y)=f(x) f(y), x, y R Put x=y=1 , we get aligned & f(2)=f(1) f(1)=9 [ f(2)=9] & f(1)^2=9 f(1)=3 & aligned Now, put x=2 and y=1 in Eq. (i), we get f(3)=f(2) f(1)=3^2 3=3^3 Now, put x=3 and y=1 in Eq. (i), we get f(4)=f(3) f(1)=3^3 3=3^4 Again, put x=4 and y=2 in Eq. (i), we get f(6)=f(4) f(2)=3^4 3^2=3^6 Alternative Method We have, aligned f(x+y) & =f(x) f(y), x, y R f(2) & =9 aligned and Now, aligned f(1+1) & =f(1) f(1) f(2) & = f(1) ^2 f(1) & = f(2) aligned Now, f(1) = f(2) f(6)=f(1+1+1+1+1+1) aligned & =f(1) f