I'm not entirely sure what you mean by "includes y=x as part of the graph" but from what I think you mean.
this function tends towards x as the second term tends towards zero so asymptotically g(x)=x
this function also tends towards x for the same reason, the second term tends to 0 as x grows.
y tends to (x-2) here as x grows so once again asymptotically y looks like x albeit offset by the constant 2.
Is this what you are after?