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: $\boxed{[1, \infty)}$

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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.