The given equation is $$y^3 + ay^2 + 2ay + 9y = \sqrt{27x}$$, and it is described as the graph of some function $$y = f(x)$$ in terms of $$x$$.

The Problem Statement:

The problem provides the equation $$y^3 + ay^2 + 2ay + 9y = \sqrt{27x}$$ and states that this represents the graph of a function $$y = f(x)$$ in terms of $$x$$.

Analysis:

• The equation involves both $$x$$ and $$y$$.
• The presence of $$\sqrt{27x}$$ implies that $$x \ge 0$$.
• The term $$y^3 + ay^2 + 2ay + 9y$$ can be factored as $$y(y^2 + ay + 2a + 9)$$.
• For this to be a function $$y = f(x)$$, for every valid input $$x$$, there must be exactly one output $$y$$.
• However, the given equation is a cubic equation in $$y$$ (or a higher-degree polynomial if $$a$$ is not a constant, but it seems to be a parameter). A cubic equation can have up to three real roots for $$y$$ for a given value of $$x$$. This generally means that $$y$$ is not a function of $$x$$ in the standard sense unless specific conditions are met for the parameter $$a$$.
• If $$a$$ is a constant, it is possible that for certain values of $$x$$, only one real value of $$y$$ satisfies the equation.
• The problem statement is likely leading to a question about finding the value of $$a$$ for which this relation defines $$y$$ as a function of $$x$$, or perhaps about the domain/range or properties of this function. Without a specific question, we can only analyze the given information.

Further Considerations:

• If the intention is for $$y$$ to be a single-valued function of $$x$$, there might be constraints on $$a$$.
• The term $$\sqrt{27x}$$ can be simplified to $$3\sqrt{3x}$$.
• The equation can be rewritten as $$y(y^2 + ay + 2a + 9) = 3\sqrt{3x}$$.

This equation defines an implicit relation between $$x$$ and $$y$$. To determine if it represents $$y$$ as a function of $$x$$, we would need to analyze the roots of the polynomial in $$y$$ for a given $$x$$. Typically, for $$y$$ to be a function of $$x$$, the equation should be solvable for $$y$$ uniquely for each $$x$$ in the domain.