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Let's analyze the problem step by step.

We have two functions, \( f(x) \) and \( g(x) \), given at selected values of \( x \). The task is to determine the transformation applied to \( f(x) \) to obtain \( g(x) \).

The values for \( f(x) \) are:
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-2 & 4 \\
0 & -3 \\
1 & -1 \\
3 & 2 \\
\end{array}
\]

The values for \( g(x) \) are:
\[
\begin{array}{c|c}
x & g(x) \\
\hline
-5 & 6 \\
-3 & -1 \\
-2 & 1 \\
0 & 4 \\
\end{array}
\]

### Step 1: Check possible horizontal shifts

Let’s try to find how \( x \) is shifted from \( f(x) \) to \( g(x) \).

- At \( x = -2 \), \( f(x) = 4 \) and \( g(x) = 6 \) corresponds to \( x = -5 \). This suggests a horizontal shift by 3 units to the left since \( -2 - 3 = -5 \).
- At \( x = 0 \), \( f(x) = -3 \) and \( g(x) = -1 \) corresponds to \( x = -3 \). This again suggests a shift of 3 units to the left because \( 0 - 3 = -3 \).
- At \( x = 1 \), \( f(x) = -1 \) and \( g(x) = 1 \) corresponds to \( x = -2 \), confirming the horizontal shift of 3 units to the left.
- At \( x = 3 \), \( f(x) = 2 \) and \( g(x) = 4 \) corresponds to \( x = 0 \), which also suggests the same horizontal shift.

Thus, the transformation includes \( x \to x + 3 \), implying the function inside the transformation is \( f(x + 3) \).

### Step 2: Check possible vertical shifts

Now let’s examine the vertical shift:

- At \( x = -2 \), \( f(-2) = 4 \) and \( g(-5) = 6 \), suggesting a vertical shift of \( +2 \) since \( 4 + 2 = 6 \).
- At \( x = 0 \), \( f(0) = -3 \) and \( g(-3) = -1 \), again a shift of \( +2 \) since \( -3 + 2 = -1 \).
- At \( x = 1 \), \( f(1) = -1 \) and \( g(-2) = 1 \), confirming a vertical shift of \( +2 \) since \( -1 + 2 = 1 \).
- At \( x = 3 \), \( f(3) = 2 \) and \( g(0) = 4 \), which also confirms a vertical shift of \( +2 \).

Thus, the vertical transformation is \( +2 \).

### Final Transformation:

The transformation from \( f(x) \) to \( g(x) \) is:
\[
g(x) = f(x + 3) + 2
\]

This matches the fourth option: 
\[
g(x) = f(x + 3) + 2
\]

### Answer: \( g(x) = f(x + 3) + 2 \)

Do you have any questions, or would you like more details on any part of this solution?

Here are five related questions you might want to explore:

1. How does horizontal shifting affect the graph of a function?
2. What is the general formula for horizontal and vertical shifts of a function?
3. How do you determine if a function undergoes a reflection or stretch in addition to shifting?
4. Can transformations include both scaling and shifting? How would that look mathematically?
5. What is the impact of combining multiple transformations like shifting, stretching, and reflecting?

**Tip**: When applying transformations, always remember that horizontal changes affect the \( x \)-coordinate, while vertical changes affect the \( y \)-coordinate!

The tables above give values of f and g at selected values of x. The graph of y = g(x) is the result of transforming the graph of f. Which of the following could be the transformation from f to g?

Understand how to transform a function by applying horizontal and vertical shifts. The problem involves determining how to shift the graph of f(x) to obtain g(x). Suitable for students in Grades 9-11.

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