Can you check my calculations please

g maps points (0,0),(1,0),(0,1) to (6,-2),(6,-1),(5,-2)

So, g(x) = (0 -1) .....(6)
..................(1 0)x + (-2)

f is a reflection through y=x-2
so, f(x) = (0 1) .....(2)
...............(1 0)x + (-2)

So; f o g = g(f(x))

= (0 -1)(0 1) ...(0 -1)(2) .....(6)
...(1 0)(1 0)x + (1 0)(-2) + (-2)

= (-1 0) ....(2) ....(6)
....(0 1)x + (2) + (-2)

= (-1 0) .....(8)
....(0 1)x + (0)

This is really stretching my brain.