Modulus function graph question

Hello, I've never encountered such a question before and I don't know how to approach it. Could you provide some pointers please?

The functions f and g are defined on the domain for all real numbers by f(x)= |x-2| and g(x)= |x|-2.

Sketch the graph of f(x) - g(x).

Student X

First, let us define each of the individual modulus functions:

|x-2| = x-2 if x≥2

         = 2-x if x<2

|x|= x if x≥0

     = -x if x<0

There are 3 critical regions, namely x<0, 0 ≤ x < 2 and x≥2

For the extreme left critical region, ie x<0,

f(x) - g(x) = |x-2| - |x| + 2 = (2-x) - (-x) +2 = 4 

In other words, you shall draw a horizontal line y=4 all the way from x=-∞ to x=0.

Thereafter, for the next critical region 0 ≤ x < 2, 

f(x) - g(x) = |x-2 - |x| + 2 = (2-x) - (x) +2 = 4-2x

In this case, you shall draw the line with equation y=4-2x for the interval between x=0 and x=2 inclusive.

I will let you figure out the final graph you need to sketch as far as the remaining critical region is concerned, which shouldn't be all too difficult if you properly understood what I have explained thus far.

Hope the above helps. Peace.

Best Regards,

Mr Koh