To rewrite the given relation as a function of $x$, we need to isolate $y$ on one side of the equation. Let's start with the given relation:

$-6y + \sqrt{x} = 3x + (x+1)^2$

Step 1: Simplify the equation

First, let's simplify the right-hand side of the equation.

$3x + (x+1)^2 = 3x + (x^2 + 2x + 1)$ $= 3x + x^2 + 2x + 1$ $= x^2 + 5x + 1$

Now the equation becomes: $-6y + \sqrt{x} = x^2 + 5x + 1$

Step 2: Isolate $y$

Now, let's move $\sqrt{x}$ to the right-hand side:

$-6y = x^2 + 5x + 1 - \sqrt{x}$

Step 3: Solve for $y$

Finally, divide both sides of the equation by $-6$ to isolate $y$:

$y = \frac{x^2 + 5x + 1 - \sqrt{x}}{-6}$

Thus, the function of $x$ is:

$y = \frac{-x^2 - 5x - 1 + \sqrt{x}}{6}$

This is the relation rewritten as a function of $x$.

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Do you want further clarification or details on any steps?

Here are 5 related questions:

1. What is the domain of the function $y(x)$?
2. How does the square root term $\sqrt{x}$ affect the behavior of the function?
3. What is the behavior of $y$ as $x \to 0$?
4. Can we simplify the expression further or analyze specific values of $x$?
5. How does the quadratic term $x^2$ influence the shape of the graph?

Tip: Always check the domain of a function when square roots or other operations with domain restrictions are involved.