Choice A is correct. The value of $k$ for which $f(k) = 3k$ can be found by substituting $k$ for $x$ and $3k$ for $f(x)$ in the given equation, $f(x) = |59 - 2x|$, which yields $3k = |59 - 2k|$. For this equation to be true, either $-3k = 59 - 2k$ or $3k = 59 - 2k$. Adding $2k$ to both sides of the equation $-3k = 59 - 2k$ yields $-k = 59$. Dividing both sides of this equation by $-1$ yields $k = -59$. To check whether $-59$ is the value of $k$, substituting $-59$ for $k$ in the equation $3k = |59 - 2k|$ yields $3(-59) = |59 - 2(-59)|$, which is equivalent to $-177 = |177|$, or $-177 = 177$, which isn't a true statement. Therefore, $-59$ isn't the value of $k$. Adding $2k$ to both sides of the equation $3k = 59 - 2k$ yields $5k = 59$. Dividing both sides of this equation by $5$ yields $k = \dfrac{59}{5}$. To check whether $\dfrac{59}{5}$ is the value of $k$, substituting $\dfrac{59}{5}$ for $k$ in the equation $3k = |59 - 2k|$ yields $3(\dfrac{59}{5}) = |59 - 2(\dfrac{59}{5})|$, which is equivalent to $\dfrac{177}{5} = |\dfrac{177}{5}|$, or $\dfrac{177}{5} = \dfrac{177}{5}$, which is a true statement. Therefore, the value of $k$ for which $f(k) = 3k$ is $\dfrac{59}{5}$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.