The problem asks for the transformation of the function $f(x) = |x|$ to a new function $g(x)$, following these transformations:

1. Horizontally stretching by a factor of 7,
2. Reflecting across the x-axis, and
3. Shifting up by 6 units.

Let's break this down step by step:

1. Horizontally stretching by a factor of 7: This means that the x-values will be scaled by $\frac{1}{7}$. So, the horizontal stretching transformation results in $f\left(\frac{x}{7}\right)$, which becomes $\left|\frac{x}{7}\right|$.

2. Reflecting across the x-axis: Reflecting the graph across the x-axis introduces a negative sign in front of the function, so the transformation becomes $-\left|\frac{x}{7}\right|$.

3. Shifting up by 6 units: To shift the graph upwards by 6 units, we add 6 to the function, resulting in $-\left|\frac{x}{7}\right| + 6$.

Thus, the final transformed function $g(x)$ is: $g(x) = -\left|\frac{x}{7}\right| + 6$

This matches the formula provided in the image: $g(x) = -\left(\frac{x}{7}\right) + 6$

Would you like further clarification on any part of this transformation process? Here are some follow-up questions you might consider:

1. What does a horizontal stretch do to the graph of a function in general?
2. How does reflecting across the x-axis affect the graph of any function?
3. Can you describe how shifting a graph up or down changes its appearance?
4. Why is the horizontal stretch by 7 represented as $\frac{x}{7}$?
5. What would happen if we shifted the graph down instead of up by 6 units?

Tip: Always apply transformations in the correct order (stretches/compressions, reflections, and then shifts).