Question

If f(x+4)=f(x) +f(4) then prove that f(0)=0, f(-4)=-f(4) and f(8)+2f(-4) =0

Answer 1

Answer:

solñ:- here,

if X=0 then,

f(0+4)= f(0)+f(4)

=> f(4)= f(0)+f(4)

=> f(0)=f(4)-f(4)

=> f(0)= 0————(1)

Again,

if X=(-4) then,

f(-4+4)=f(-4)+f(4)

=> f(0)= f(-4)+f(4)

=> 0= f(-4)+f(4) [from 1, f(0)=0]

=>f(-4)= -f(4)——––(2)

Now,

if X=4 then,

f(4+4)= f(4)+f(4)

=>f(8)=2.f(4)

=>f(8)-2.f(4)=0

=>f(8)+[-2.f(4)]=0

=>f(8)+2.[-f(4)]=0

=>f(8)+2.f(-4)=0 [from 2, -f(4)=f(-4)]

Hence proved...