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To find the inverse of the function \( f(x) = (2x + 1)^2 \) for \( x \geq -\frac{1}{2} \), let's proceed step by step:

### Step 1: Set \( y = f(x) \)
We start by setting \( y = (2x + 1)^2 \).

\[
y = (2x + 1)^2
\]

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \).

1. Take the square root of both sides:
   \[
   \sqrt{y} = 2x + 1
   \]
   Since \( x \geq -\frac{1}{2} \), we will only take the positive square root.

2. Solve for \( x \):
   \[
   2x = \sqrt{y} - 1
   \]
   \[
   x = \frac{\sqrt{y} - 1}{2}
   \]

### Step 3: Replace \( y \) with \( x \)
The inverse function \( f^{-1}(x) \) is given by:
\[
f^{-1}(x) = \frac{\sqrt{x} - 1}{2}
\]

### Domain of the Inverse Function
Since the original function \( f(x) = (2x + 1)^2 \) is defined for \( x \geq -\frac{1}{2} \), the range of \( f(x) \) starts from \( (2(-\frac{1}{2}) + 1)^2 = 0 \). Therefore, the inverse function is defined for \( x \geq 0 \).

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{\sqrt{x} - 1}{2}, \quad x \geq 0
\]

Would you like more details or further clarification?

### Related Questions:
1. What are the general steps to find the inverse of a quadratic function?
2. How does the domain restriction \( x \geq -\frac{1}{2} \) affect the inverse function?
3. Can you explain why we take the positive square root when solving for \( x \)?
4. What is the geometric interpretation of a function and its inverse on a graph?
5. How do we verify if a function and its inverse are correct by composition?

### Tip:
To confirm whether two functions are inverses, check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

What is the inverse of f(x) = (2x + 1)^2 for x ≥ -1/2?

Discover how to find the inverse of the quadratic function f(x) = (2x + 1)^2 for x ≥ -1/2. This step-by-step guide explains the process of solving for the inverse function, including handling domain restrictions and solving for x.

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