The problem describes a transformation of the parent function $f(x) = |x|$ to a transformed function $g(x)$. We need to identify the transformations, write the function $g(x)$ in function notation and as an equation, and determine the domain and range of $g(x)$.

1. Problem Description

The problem describes a transformation of the parent function $f(x) = |x|$ to a transformed function $g(x)$. We need to identify the transformations, write the function $g(x)$ in function notation and as an equation, and determine the domain and range of $g(x)$.

2. Solution Steps

First we identify the transformations.

The graph of $g(x)$ is the same shape as the graph of $f(x)$, but it is shifted horizontally and vertically. The vertex of $f(x)$ is at (0,0). The vertex of $g(x)$ is at (1,2).

Therefore, there are horizontal and vertical shifts. There are no stretches or compressions.

* Horizontal Shift: The vertex of $g(x)$ is at $x=1$, so there is a horizontal shift of 1 unit to the right.

* Vertical Shift: The vertex of $g(x)$ is at $y=2$, so there is a vertical shift of 2 units up.

The transformed function can be written as $g(x) = f(x-h) + k$, where $h$ is the horizontal shift and $k$ is the vertical shift. In our case, $h=1$ and $k=2$.

Thus, $g(x) = |x-1| + 2$.

Now let's determine the domain and range.

* Domain: The domain of $f(x) = |x|$ is all real numbers, i.e. $(-\infty, \infty)$. Horizontal and vertical shifts do not change the domain.

So, the domain of $g(x)$ is also $(-\infty, \infty)$.

* Range: The range of $f(x) = |x|$ is $[0, \infty)$. The vertical shift of $g(x)$ changes the range. The range of $g(x)$ is $[2, \infty)$ since the vertex is at $y=2$.

3. Final Answer

1) Horizontal Shifts and Vertical Shifts. (There are no Stretches/Compressions).

2) Vertical Shifts

3) Horizontal Shifts

4) $g(x) = |x-1| + 2$

5) $g(x) = |x-1| + 2$

6) Domain of $g(x)$: $(-\infty, \infty)$

7) Range of $g(x)$: $[2, \infty)$