We are tasked with finding the maximum and minimum values of the function:

\[
f(x) = 2|x+2| + 1
\]

### Step 1: Analyze the Absolute Value
The expression contains an absolute value term, \( |x+2| \). This splits the function into two cases depending on whether \( x + 2 \) is positive or negative:

- When \( x + 2 \geq 0 \), meaning \( x \geq -2 \), the absolute value \( |x+2| = x+2 \).
- When \( x + 2 < 0 \), meaning \( x < -2 \), the absolute value \( |x+2| = -(x+2) = -x - 2 \).

### Step 2: Write the Piecewise Function
Now we can write the function \( f(x) \) without absolute values:

\[
f(x) = 
\begin{cases} 
2(x+2) + 1 = 2x + 5, & \text{if } x \geq -2 \\
2(-x - 2) + 1 = -2x - 3, & \text{if } x < -2
\end{cases}
\]

### Step 3: Analyze Each Piece
1. **For \( x \geq -2 \):**
   - The function is \( f(x) = 2x + 5 \), which is a **linear increasing function**.

2. **For \( x < -2 \):**
   - The function is \( f(x) = -2x - 3 \), which is a **linear decreasing function**.

### Step 4: Check for Continuity at \( x = -2 \)
To ensure the function is continuous, let's evaluate \( f(x) \) at \( x = -2 \) for both cases:
- When \( x = -2 \), for \( x \geq -2 \), we have:
  \[
  f(-2) = 2(-2) + 5 = 1
  \]
- When \( x = -2 \), for \( x < -2 \), we also have:
  \[
  f(-2) = -2(-2) - 3 = 1
  \]
  
So, the function is continuous at \( x = -2 \).

### Step 5: Determine Maximum and Minimum
- As \( x \to -\infty \), the function \( -2x - 3 \to +\infty \), meaning the function **increases indefinitely** as \( x \) becomes more negative.
- As \( x \to +\infty \), the function \( 2x + 5 \to +\infty \), meaning the function also **increases indefinitely** for large positive \( x \).

However, the function has a **minimum value** at \( x = -2 \), where \( f(-2) = 1 \).

### Conclusion:
- The **minimum value** of the function is \( f(x) = 1 \) at \( x = -2 \).
- The function has **no maximum** because it increases without bound as \( x \to \pm \infty \).

Would you like further details or have any questions?

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Here are some related questions you might consider:
1. What happens to the function if we change the coefficient of \( |x+2| \)?
2. How would you find the roots of \( f(x) = 0 \)?
3. Can the function ever be negative? Why or why not?
4. How would the graph of \( f(x) \) change if it were \( f(x) = 2|x-2| + 1 \)?
5. How does the absolute value affect the symmetry of the function?

**Tip:** When dealing with absolute value functions, always break the function into pieces to handle different cases.

Learn how to find the maximum and minimum of the function f(x) = 2|x+2| + 1 by analyzing the absolute value and breaking the function into piecewise components. This problem involves key algebraic concepts such as continuity and linear functions. Suitable for students in Grades 9-12.

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