Modulus function graph question

Hello, I've never encountered this sort of question before and I don't know how to approach it. Could you explain it please? 

The functions f and g are defined on the domain of 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 from x=0 to x=2.

I shall let you figure out the graph you need to sketch for the remaining critical region, which shouldn't be all too difficult if you understood what I have explained thus far.

Hope it helps. Peace.

Best Regards,

Mr Koh