If `f(x)=x+abs(x)+1` , then which of the follow represents `f(x)`

Options

A. `f(x)={(2x-1, xge0),(1,xlt0):}`
B. `f(x)={(2x+1, x ge0),(1,x lt0):}
C. `f(x)={(2x, xle0),(2,xgt0):}
D. `f(x)={(2x+1, xle0),(1,xgt0):}