We are given the function \( f(x) = 2|x+2| + 1 \). We need to determine the domain and range of this function.

### **Domain:**
The domain refers to the set of all possible input values \(x\) for which the function is defined.

- The absolute value function \( |x + 2| \) is defined for all real numbers. 
- Since there are no restrictions (like division by zero or square roots of negative numbers), \( f(x) \) is defined for all real values of \(x\).

Thus, the **domain** is:
\[
\boxed{(-\infty, \infty)}
\]

### **Range:**
The range refers to the set of all possible output values of the function.

- The absolute value function \( |x+2| \) is always non-negative, meaning \( |x+2| \geq 0 \).
- Multiplying by 2, we have \( 2|x+2| \geq 0 \).
- Adding 1 to the expression gives \( 2|x+2| + 1 \geq 1 \).

Thus, the minimum value of the function occurs when \( x = -2 \), which gives \( f(x) = 2(0) + 1 = 1 \). As \( x \) moves away from \(-2\), the value of \( f(x) \) increases without bound.

Therefore, the **range** is:
\[
\boxed{[1, \infty)}
\]

---

Do you want more details or have any questions? Here are 5 related questions to explore further:

1. What is the vertex of the function \( f(x) = 2|x+2| + 1 \)?
2. How do we graph absolute value functions like \( f(x) \)?
3. How does modifying the constant inside the absolute value affect the domain and range?
4. Can the function \( f(x) = 2|x+2| + 1 \) have a horizontal shift? If so, how?
5. How does the multiplier (like 2 in this case) affect the slope of the graph of the function?

**Tip:** When determining the range of a function, consider how the function behaves for both small and large values of \(x\), especially for absolute value functions that reflect symmetry.

Discover the domain and range of the absolute value function f(x) = 2|x+2| + 1. This problem explores key algebraic concepts, including the absolute value function, and is suitable for high school students (Grades 9-12).

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