https://file.aiquickdraw.com/mathbot/file/1726776427750bg1s07db.jpg

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.

The function f(x) is given by f(x) = |x + 2| - 5. Which of the statements are true? Select TWO correct answers.

Explore the function f(x) = |x + 2| - 5 and learn about its domain, range, graph characteristics, and symmetry. This Grade 11 Algebra II problem demonstrates key mathematical concepts involving absolute value functions and their transformations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation