已知函数f（x）=|x﹣1|，则下列函数与f（x）相等的函数是（   ）

A：g（x）= $\frac{\left|x^2-1\right|}{\left|x+1\right|}$

B：g（x）= $\{\begin{array}{c}\frac{\left|x^2-1\right|}{\left|x+1\right|},x\ne -1\\ 2,x=-1\end{array}$

C：g（x）= $\{\begin{array}{c}x-1,x>0\\ 1-x,x\le 0\end{array}$

D：g（x）=x﹣1