To find the range of the function $f(x) = -|x + 2| - 2$, we can analyze the behavior of the function step-by-step and use graphing tools to confirm the result.

1. Understand the function structure:

• The absolute value function $|x + 2|$ reflects values symmetrically around $x = -2$. It is piecewise and has a "V" shape.
• The negative sign in front of the absolute value $-|x + 2|$ inverts the "V" shape, turning it into a downward-facing "V".
• Finally, subtracting 2 from the entire expression $-|x + 2| - 2$ shifts the graph 2 units downward.

2. Vertex of the function:

• The vertex of the function occurs where $|x + 2| = 0$, i.e., when $x = -2$.
• Substituting $x = -2$ into the function, we get: $f(-2) = -| -2 + 2 | - 2 = -0 - 2 = -2$ The vertex is at $(-2, -2)$.

3. Behavior of the function:

• As $x$ moves away from $-2$, the value of $|x + 2|$ increases. Since the absolute value is multiplied by -1, the function decreases.
• The function has no upper bound, since it is always decreasing as we move away from $x = -2$.

4. Range of the function:

• The maximum value of $f(x)$ occurs at the vertex, which is $-2$.
• The function decreases without bound as $x$ moves away from the vertex.
• Therefore, the range of the function is: $(-\infty, -2]$

If you were to graph this function using a graphing tool, you'd observe a downward-facing "V" with a maximum at $(-2, -2)$ and the rest of the graph extending downward.

Would you like more details or have any questions?

Related Questions:

1. How would the graph change if the absolute value term were $|x - 2|$ instead of $|x + 2|$?
2. What is the domain of the function $f(x) = -|x + 2| - 2$?
3. How would the range change if we added a constant $c$ to the function $f(x) = -|x + 2| - 2 + c$?
4. Can you find the derivative of $f(x) = -|x + 2| - 2$, and what does it tell us about the function's behavior?
5. How would the function behave if the absolute value term were multiplied by a positive constant (e.g., $-2|x + 2| - 2$)?

Tip:

Always check the vertex of an absolute value function when looking for the maximum or minimum values; it helps in determining the range.