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To solve this problem, we need to identify the graph of \( f(-x) \) based on the given graph of \( f(x) = x + 1 \).

### Key Concepts:
- The function \( f(x) = x + 1 \) is a linear function with a slope of 1 and a y-intercept of 1.
- The graph of \( f(-x) \) reflects the graph of \( f(x) \) across the y-axis. This is because replacing \( x \) with \( -x \) inverts the sign of the x-values.

### Steps to Identify the Graph of \( f(-x) \):
1. **Original Graph of \( f(x) = x + 1 \):**
   - The given graph of \( f(x) = x + 1 \) is a straight line with a slope of 1, meaning it rises 1 unit for every 1 unit it moves to the right.

2. **Reflection for \( f(-x) \):**
   - Reflecting the graph across the y-axis means every point \( (x, y) \) on the graph of \( f(x) \) will be mapped to \( (-x, y) \) for the graph of \( f(-x) \).

3. **Examining the Options:**
   - We need to compare the options and find the graph that mirrors \( f(x) = x + 1 \) across the y-axis.

### Analyzing the Options:
- **Option 1** (blue dotted line) appears to have a positive slope and does not seem to match the reflection of the original function.
- **Option 2** (black dashed line) has a negative slope and seems to be a reflection of the original line.
- **Option 3** (green dotted line) has a positive slope, but it is less steep than the original function, so it’s not a correct reflection.
- **Option 4** (orange dashed line) also has a negative slope and seems to match the reflection, but visually, it’s less steep than Option 2.

### Conclusion:
The correct answer is **Option 2** (the black dashed line), which is the reflection of the original function \( f(x) = x + 1 \) across the y-axis.

### Final Answer:
- The graph of \( f(-x) \) is **Option 2**.

Would you like more details on why this is the correct reflection, or do you have any further questions?

Here are some related questions to deepen your understanding:
1. How do you find the equation of a function reflected over the y-axis?
2. What happens when you reflect a quadratic function across the y-axis?
3. How does the graph of \( f(x) \) change when reflected over the x-axis instead?
4. How does shifting a graph horizontally affect the function's equation?
5. Can you reflect non-linear functions in the same way as linear ones?

**Tip:** When reflecting functions over the y-axis, always invert the sign of the x-values in the function's formula.

Given the graph of f(x) = x + 1, which option is the graph of f(-x)?

Explore how the graph of f(x) = x + 1 reflects across the y-axis to produce the graph of f(-x). This problem focuses on function transformations and graph reflections, suitable for students in Grades 9-11.

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