SOLUTION: Is f (x+3) same with f(×) + f(3) show solution with explanation

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Question 1085052: Is f (x+3) same with f(×) + f(3)
show solution with explanation
Found 2 solutions by Fombitz, natolino_2017:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Sometimes.
f%28x%2B3%29=f%28x%29%2Bf%283%29
When f%28x%29=0 then,
f%280%2B3%29=f%280%29%2Bf%283%29
f%283%29=f%280%29%2Bf%283%29
0=0%2B0
0=0
True.
.
.
.
Sometimes not,
f%28x%2B3%29=f%28x%29%2Bf%283%29
When f%28x%29=x%5E2
%28x%2B3%29%5E2=x%5E2%2B9
x%5E2%2B6x%2B9=x%5E2%2B9
6x=0
False.

Answer by natolino_2017(77) About Me  (Show Source):
You can put this solution on YOUR website!
f(x+3) = f(x) + f(3), it means that f is a linear transformation.
so if f(x) = ax +b,
f(x+3) = a(x+3) + b = ax +3a +b
f(3) = 3a + b.
So if f(x+3) = f(x) + f(3)
ax+3a+b = ax +b +3a +b
So b=0, and a any real number.
Answer: f(x) = ax with a any real number.
@natolino_