Answer:f(1) = f(3) = f(9) = f(27) = f(81).
Step-by-step explanation:To show that f(1) = f(3) = f(9) = f(27) = f(81), we can use the given equation and substitute x with different values.
Let x = 1:
f(3x) = f(3) + f(x)
f(3(1)) = f(3) + f(1)
f(3) = f(3) + f(1) (since f(3x) = f(3) + f(x) when x = 1)
f(1) = 0
Now, let x = 3:
f(3x) = f(3) + f(x)
f(3(3)) = f(3) + f(3)
f(9) = 2f(3)
Since we already know that f(3) = f(3) + f(1), we can substitute it into the equation above:
f(9) = 2f(3) = 2(f(3) + f(1)) = 2f(3) + 2f(1)
f(9) - 2f(3) = 2f(1)
Now let x = 9:
f(3x) = f(3) + f(x)
f(3(9)) = f(3) + f(9)
f(27) = f(3) + f(9)
We can substitute the expression for f(9) we just derived:
f(27) = f(3) + (f(9) - 2f(3))
f(27) = f(9) - f(3)
f(27) = 2f(3) - f(3)
f(27) = f(3)
Finally, let x = 27:
f(3x) = f(3) + f(x)
f(3(27)) = f(3) + f(27)
f(81) = f(3) + f(27)
We can substitute the expression for f(27) we just derived:
f(81) = f(3) + f(27)
f(81) = f(3) + f(3)
f(81) = 2f(3)
Since we know that f(27) = f(3), we can substitute it into the equation above:
f(81) = 2f(3) = 2(f(27)) = 2(f(9) - f(3)) = 2f(9) - 4f(3)
We also know that f(9) = 2f(3) + 2f(1) from our earlier work, so we can substitute it into the equation above:
f(81) = 2f(9) - 4f(3) = 2(2f(3) + 2f(1)) - 4f(3) = 4f(3) + 4f(1) - 4f(3) = 4f(1)
Therefore, we have shown that f(1) = f(3) = f(9) = f(27) = f(81).