Let's analyze the problem and the data in the image step by step.

Given Data:

We have two tables:

1. Table for $f(x)$:

2. Table for $g(x)$:

We are asked to find which transformation of $f(x)$ results in $g(x)$. The possible transformations are:

• (A) $g(x) = f(x - 4) - 1$
• (B) $g(x) = f(x - 3) + 1$
• (C) $g(x) = f(x + 2) - 5$
• (D) $g(x) = f(x + 1) + 7$

Step-by-Step Analysis:

The goal is to determine how $g(x)$ is related to $f(x)$ by comparing the two tables.

Check for Horizontal Shifts:

We observe the corresponding values of $f(x)$ and $g(x)$. For example:

• When $x = 0$, $f(0) = 2$, but $g(0) = 0$.
• When $x = 4$, $f(4)$ is not directly given, but we can see that at $x = 2$, $f(2) = 3$ and $g(4) = 1$.

This suggests a shift in $x$.

Check Transformation (A):

For $g(x) = f(x - 4) - 1$, we shift the input of $f(x)$ to the right by 4 and then subtract 1 from the output.

Let’s test this:

• For $x = 0$, $f(0 - 4) = f(-4) = 1$, and $1 - 1 = 0$, which matches $g(0)$.
• For $x = 4$, $f(4 - 4) = f(0) = 2$, and $2 - 1 = 1$, which matches $g(4)$.
• For $x = 6$, $f(6 - 4) = f(2) = 3$, and $3 - 1 = 2$, which matches $g(6)$.
• For $x = 9$, $f(9 - 4) = f(5) = 4$, and $4 - 1 = 3$, which matches $g(9)$.

Thus, transformation (A) works.

Conclusion:

The correct transformation is (A) $g(x) = f(x - 4) - 1$.

Do you need further clarification or details on any part of this problem? Here are some related questions for practice:

1. How would the transformation change if we added a vertical shift instead of a horizontal one?
2. What is the effect of applying $g(x) = f(x + 3)$ to $f(x)$?
3. Can you identify how a reflection over the x-axis would change $f(x)$?
4. How do horizontal shifts affect the graph of a function?
5. How would $g(x) = f(-x) + c$ transform $f(x)$?

Tip: Always try applying transformations to a few points to check if they match the given function!