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:

\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$$. Would you like further details or have any questions? --- Here are some related questions you might consider: 1. What happens to the function if we change the coefficient of $$|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.