The function given is $f(x) = |x + 2| - 5$, and you are asked to select two correct statements about this function. Let's break down its characteristics:

1. Domain of the Function:

The function $f(x) = |x + 2| - 5$ involves an absolute value, which is defined for all real numbers. Therefore, the domain of the function is all real numbers, or $(-\infty, \infty)$.

2. Range of the Function:

The absolute value expression $|x + 2|$ is always non-negative, so the minimum value it can take is 0 (when $x = -2$). When $x = -2$, the value of the function is: $f(-2) = |(-2) + 2| - 5 = 0 - 5 = -5$ Since the absolute value can grow indefinitely as $|x + 2|$ increases, the range of the function is $[-5, \infty)$.

3. Graph Characteristics:

• The graph of $f(x) = |x + 2| - 5$ is a V-shaped graph, as it is a transformation of the standard absolute value function.
• The graph is symmetric about $x = -2$, because the absolute value function creates symmetry around the point where the expression inside the absolute value is zero.

4. Vertex of the Graph:

The vertex of the graph occurs at $x = -2$, where the function reaches its minimum value $-5$. Therefore, the graph has a minimum point at $(-2, -5)$.

5. Symmetry of the Graph:

As mentioned earlier, the graph is symmetric around the line $x = -2$.

Summary of Possible Correct Statements:

• Domain: All real numbers.
• Range: $y \geq -5$.
• Graph: Has a minimum at $(-2, -5)$ and is symmetric about $x = -2$.

Would you like more details about any of these points or clarifications? Here are 5 related questions to explore further:

1. How do transformations affect the graph of an absolute value function?
2. What are the properties of the absolute value function $|x|$?
3. How can we determine the vertex of an absolute value function?
4. What is the range of a transformed absolute value function?
5. How do we algebraically verify the domain of any given function?

Tip: Always look at how shifting the graph of a parent function (like $|x|$) affects its domain, range, and symmetry.