The given equation is:

$y + \sqrt{x} = -4x + (x - 4)^2$

Step 1: Solve for $y$

We will first isolate $y$ on one side.

$y = -4x + (x - 4)^2 - \sqrt{x}$

This represents a function $y = f(x)$.

Step 2: Determine the domain of the function

To find the domain, we need to consider the restrictions on $x$ that come from the square root, as well as any other restrictions.

• Square Root Restriction: The expression $\sqrt{x}$ implies that $x$ must be non-negative because the square root of a negative number is not defined in the real number system. $x \geq 0$

There are no other restrictions in the function, since the rest of the expression consists of polynomials, which are defined for all real numbers.

Step 3: Final Domain

The domain is the set of $x$-values for which the function is defined. Since $x$ must be non-negative, the domain is:

$[0, \infty)$

Final Answer:

The domain of the function is $[0, \infty)$.

Would you like more details or have any questions?

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Here are 5 questions to expand on this topic:

1. How do you handle square roots in function domains?
2. What happens if a square root is in the denominator?
3. How do polynomials affect the domain of a function?
4. What would happen to the domain if we had $\sqrt{x - 3}$ instead of $\sqrt{x}$?
5. How do you graph a function with a restricted domain?

Tip: Always check for square roots and denominators when determining the domain of a function.