Nonlinear functions SAT question

Choice D is correct. If the graph of $y=g(x)$ is the result of shifting the graph of $y=f(x)$ down $k$ units in the xy-plane, the function $g$ can be defined by an equation of the form $g(x)=f(x)-k$. It’s given that $f(x)=7x^3$ and the graph of $y=g(x)$ is the result of shifting the graph of $y=f(x)$ down $2$ units. Substituting $7x^3$ for $f(x)$ and $2$ for $k$ in the equation $g(x)=f(x)-k$ yields $g(x)=7x^3-2$. 

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This equation defines a function $g$ for which the graph of $y=g(x)$ is the result of shifting the graph of $y=f(x)$ up, not down, $2$ units.