consider two cases, viz x>=3 and x<3.
now check whether f(x) is bijection. if it is then inverse exists.
after that find the inverse of each "piece".
sketching the graph may also help.
For (I don't see any reason to include the " ")
x- 3< 0 so |x- 3|= -(x-3). Then f(x)= 2x- 3- |x- 3|= 2x- 3+ (x- 3)= 3x- 6 so that as chi-sigma said.
For , |x- 3|= x- 3 so that
f(x)= 2x - 3-| x - 3 |= 2x- 3- (x- 3)= x which has inverse .
Of course, neither of those answers the question "what is the inverse of ?"