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### Given Function:
\[ f(x) = (x + 4)^2 - 5 \]

#### 1. **Domain (D):**
The function \( f(x) = (x + 4)^2 - 5 \) is a quadratic function. Since there are no restrictions (like division by zero or square roots of negative numbers), the domain of any quadratic function is all real numbers.

**Domain:** 
\[ D: (-\infty, \infty) \]

#### 2. **Range (R):**
To find the range, we analyze the vertex form of the quadratic function. The function is written as:
\[ f(x) = (x + 4)^2 - 5 \]

This is a parabola that opens upwards (because the coefficient of the quadratic term is positive). The vertex occurs at \( x = -4 \), and the minimum value of the function occurs at the vertex. Substituting \( x = -4 \) into the function:
\[
f(-4) = (-4 + 4)^2 - 5 = 0 - 5 = -5
\]
Thus, the minimum value of \( f(x) \) is \( -5 \). Since the parabola opens upwards, the function's values will increase from \( -5 \) to infinity.

**Range:**
\[ R: [-5, \infty) \]

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Would you like more details on this solution? Here are 5 related questions to expand your understanding:

1. How do you convert a general quadratic function to vertex form?
2. Why does the range of the function start from the minimum value of \( -5 \)?
3. What would change in the domain or range if the quadratic function opened downwards?
4. Can the range of a quadratic function ever include negative infinity?
5. How would transformations affect the domain and range of a quadratic function?

**Tip:** The domain of most polynomial functions (like quadratics) is always all real numbers unless specific restrictions are involved, such as square roots or division by zero.

Find the domain and range of the function f(x) = (x + 4)^2 - 5. Explain how you know.

Find the domain and range of the quadratic function f(x) = (x + 4)^2 - 5. Learn how to determine the domain as all real numbers and the range starting from -5. Suitable for Grades 8-10.

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