To find the range of the function \[ f(x) = \frac{(x - 2)^2}{4x^2 + 16x}, \] we can follow these steps:

### Step 1: Simplify the function
We first simplify the denominator:

\[
4x^2 + 16x = 4x(x + 4).
\]

So the function becomes:

\[
f(x) = \frac{(x - 2)^2}{4x(x + 4)}.
\]

### Step 2: Analyze the domain
The denominator \( 4x(x + 4) \) cannot be zero, so we exclude the values \( x = 0 \) and \( x = -4 \) from the domain. Thus, the domain is all real numbers except \( x = 0 \) and \( x = -4 \).

### Step 3: Determine behavior at the critical points
Now, let's analyze the behavior of \( f(x) \) for large values of \( x \), and near the undefined points \( x = 0 \) and \( x = -4 \).

1. **As \( x \to \infty \) or \( x \to -\infty \):**
   For large values of \( x \), the highest power terms dominate, and we get:

   \[
   f(x) \approx \frac{x^2}{4x^2} = \frac{1}{4}.
   \]

   So, \( f(x) \) approaches \( \frac{1}{4} \) as \( x \to \infty \) or \( x \to -\infty \).

2. **At \( x = 0 \) and \( x = -4 \):**
   The function is undefined at these points.

3. **As \( x \to -4 \) from either side:**
   The denominator approaches zero and the numerator is finite, so \( f(x) \to \infty \) or \( f(x) \to -\infty \).

### Step 4: Check for minimum values
Since \( (x - 2)^2 \geq 0 \), the numerator is always non-negative. Thus, \( f(x) \geq 0 \).

At \( x = 2 \), the numerator is zero, so \( f(2) = 0 \).

### Step 5: Conclusion about the range
The function approaches \( 0 \) as the minimum value at \( x = 2 \) and approaches \( \frac{1}{4} \) for large \( x \). The function also becomes unbounded near \( x = -4 \). Hence, the range of the function is:

\[
\boxed{[0, \infty)}.
\]

Would you like a more detailed breakdown or have any questions?

Here are some related questions you might find helpful:
1. How do you find the domain of a rational function?
2. What happens to a function at vertical asymptotes?
3. How can you determine if a function has a horizontal asymptote?
4. How do you find the critical points of a function?
5. What methods can be used to check the limits of a function?

**Tip:** Always check for vertical and horizontal asymptotes when working with rational functions.

Learn how to find the range of the rational function (x - 2)^2 / (4x^2 + 16x) through step-by-step analysis, including simplification, domain restrictions, and asymptotic behavior. Suitable for students in Grades 10-12.

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