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.