The answer is
[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
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Here's how I got that answer:
Start with the piecewise definition for y = |x|.
[tex]g(x) = \begin{cases}x \text{ if }x \ge 0 \\ -x \text{ if }x < 0\end{cases}[/tex]
Everywhere you see an 'x', replace it with x+2
[tex]g(x+2) = \begin{cases}x+2 \text{ if }x+2 \ge 0 \\ -(x+2) \text{ if }x+2 < 0\end{cases}[/tex]
[tex]g(x+2) = \begin{cases}x+2 \text{ if }x \ge -2 \\ -x-2 \text{ if }x < -2\end{cases}[/tex]
Now tack on "-4" at the end of each piece so that we shift the function down 4 units
[tex]g(x+2)-4 = \begin{cases}x+2-4 \text{ if }x \ge -2 \\ -x-2-4 \text{ if }x < -2\end{cases}[/tex]
[tex]g(x+2)-4 = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
Check out the attached images below. In figure 1, I graph y = x-2 and y = -x-6 as separate equations on the same xy coordinate system. Then in figure 2, I combine them to form the familiar V shape you see with any absolute value graph.
