To determine which function matches $g(x)$ based on the graph provided:

1. Identify the transformation: The graph of $g(x)$ is a transformation of the graph of $f(x)$.

2. Analyze the transformations:

   • Horizontal Shift: If $g(x) = f(x - a)$, the graph of $g(x)$ would be a horizontal shift of $f(x)$ to the right by $a$ units.
   • Vertical Stretch: If $g(x) = 2f(x)$, the graph of $g(x)$ would be a vertical stretch of $f(x)$ by a factor of 2.

   By analyzing the graph:

   • Horizontal Shift: The graph of $g(x)$ appears to be shifted to the right by 2 units compared to $f(x)$.
   • Vertical Stretch: The graph of $g(x)$ appears to be a vertical stretch of $f(x)$ by a factor of 2.

3. Match the transformations:

   • The correct transformation that matches both a shift to the right by 2 units and a vertical stretch by a factor of 2 is option D: $g(x) = 2f(x - 2)$.

Thus, the correct answer is D.

Would you like more details or have any further questions?

Here are five related questions for further exploration:

1. How does a horizontal shift affect the equation of a function?
2. What effect does a vertical stretch have on the graph of a function?
3. Can you explain the difference between horizontal and vertical transformations in functions?
4. How would the equation change if $g(x)$ was a reflection of $f(x)$?
5. What is the effect of a vertical shift on a function's graph?

Tip: When dealing with transformations of functions, always pay close attention to the direction and magnitude of shifts and stretches, as they directly affect the function's equation.