KöMaL 2005 B3851

Consider the functions obtained as compositions of the functions $f\colon x \mapsto \frac{1}{x}$ , $g\colon x \mapsto 1-\frac{1}{x}$ , each used a finite number of times; for example $f\circ g\circ f\colon x \overset{f}\mapsto \frac{1}{x} \overset{g}\mapsto 1- \dfrac{1}{\frac{1}{x}} = 1- x \overset{f}\mapsto \frac{1}{1-x}$ . Is it possible to obtain the function $x \mapsto x$+1 in this way? (It is allowed to cancel finitely many numbers from its domain.)