Consider the following relation.

−6y+x‾‾√=3x+(x+1)2−6y+x=3x+(x+1)^2

Step 1 of 3 :

Rewrite the relation as a function of x.

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 \( x \).

---

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.

Consider the following relation: −6y + √x = 3x + (x + 1)^2. Rewrite the relation as a function of x.

Learn how to rewrite the equation −6y + √x = 3x + (x + 1)^2 as a function of x. This step-by-step solution covers algebraic concepts, including square roots, quadratic expansion, and function isolation. Suitable for students in Grades 9-12.

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