To write the function $g(x)$ based on the transformation of the graph of $f(x)$, we need to analyze how the graph of $f$ has been transformed to create the graph of $g$.

Step-by-step analysis:

1. Original function $f(x)$: The function $f(x)$ appears to be a V-shaped graph, which is typical of the absolute value function. Based on the image, the graph of $f(x)$ seems to be: $f(x) = |x|$ since the graph has a vertex at the origin $(0, 0)$ and slopes of 1 and -1.

2. Transformation of $f(x)$ to $g(x)$:

   • The graph of $g(x)$ is the result of reflecting $f(x)$ across the x-axis, as the V-shape is inverted.
   • Additionally, the graph of $g(x)$ is shifted downward by 2 units.

Conclusion:

The transformation can be described as a vertical reflection and a downward shift. This transformation can be written as: $g(x) = -f(x) - 2$

Since $f(x) = |x|$, substituting this in gives: $g(x) = -|x| - 2$

Thus, the function $g(x)$ is: $g(x) = -|x| - 2$

Would you like further clarification or details on this solution?

Here are five related questions to explore:

1. What are other common transformations of absolute value functions?
2. How does a horizontal shift affect the equation of an absolute value function?
3. What is the general formula for reflecting a function across the x-axis?
4. How does vertical stretching or compression affect the graph of a function?
5. Can you provide an example of a horizontal reflection and how that changes the function?

Tip: Always pay attention to whether a transformation involves shifts, reflections, or stretches/compressions to properly adjust the function's equation.