Consider the following relation.

y+x‾‾√=−4x+(x−4)2y+x=−4x+(x−4)^2

Step 3 of 3 :  

Determine the implied domain of the function found in the first step. Express your answer in interval notation.

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.

Consider the following relation: y + sqrt(x) = −4x + (x − 4)^2. Determine the implied domain of the function found in the first step. Express your answer in interval notation.

This problem explores how to find the domain of a function involving a square root. The equation y = −4x + (x − 4)^2 - sqrt(x) highlights key concepts in algebra and function domains. Learn how square roots impact the domain and understand how to express it in interval notation. Suitable for Grades 9-12 students.

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