Since $\log _{3}4 > 0$, we use the first part of the piecewise function:
$$f(x) = 3^{x}$$
Thus,
$$f(\log _{3}4) = 3^{\log _{3}4}$$
Using the property of logarithms that says $a^{\log_a{n}} = n$, we get:
$$f(\log _{3}4) = 3^{\log _{3}4} = 4$$
Therefore, the answer is $\boxed{4}$.