To apply the transformation to the given function $f(x) = |x+1| - 1$ with a vertical stretch by a factor of 3, we multiply the entire function by 3. A vertical stretch affects the output (the $y$-values) of the function.

So, to write the new function $g(x)$, apply the vertical stretch:

$g(x) = 3 \cdot f(x)$

Since $f(x) = |x+1| - 1$, the new function becomes:

$g(x) = 3 \cdot (|x+1| - 1)$

Now, distribute the 3:

$g(x) = 3 \cdot |x+1| - 3$

Thus, the transformed function is:

$g(x) = 3|x+1| - 3$

Final answer:

$g(x) = 3|x+1| - 3$

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Would you like further clarification, or do you have any questions about this transformation?

Here are five related questions:

1. What would the transformation look like with a vertical shrink instead of a stretch?
2. How does a horizontal translation affect the function?
3. What happens if we apply both horizontal and vertical transformations?
4. Can you graph the original and transformed functions for comparison?
5. How does the shape of the graph change with different absolute value expressions?

Tip: When applying vertical transformations, always adjust the output of the function (i.e., multiply the whole function by the stretch or shrink factor).