The domain of the sine function is all of the reals. With the natural log, you have to restrict x so that the argument of the log is positive. But note that for all x, -1 ≤ sin(x) ≤ 1. So inside the log
2 ≤ 3 + sin(x) ≤ 4.
That is, 3 + sin(x) is always positive. The domain of f is therefore all real numbers
domain = (-∞, ∞)
As for the range, the above shows that the term inside of the log can take on every value from 2 to 4 including the end points. So the range is all values from ln(2) to ln(4).
range = [ln(2), ln(4)]
The function is periodic with period 2π because the sine is periodic with period 2π. The sine function has the property
sin(x + 2π) = sin(x) for all real x.
And 2π is the smallest value for which this is true. Then we see that
f(x + 2π) = ln(3 + sin(x + 2π)) = ln(3 + sin(x)) = f(x).
f is neither even nor odd. The sine function itself is odd, but the constant function 3 is even. The sum 3 + sin(x) is neither. The log by itself has no symmetry, though that doesn't preclude it being involved in a symmetric function. Note that
f(-x) = ln(3 + sin(-x)) = ln(3 - sin(x))
which is not the same as f(x) or - f(x).