Let . Then . Hence g is a strictly increasing function. A strictly increasing function can only cross the x-axis at most once.
Suppose doesn’t cross the x-axis at all. The only way this is possible for a strictly increasing differentiable (and thus continuous) function is if either (i) is negative for all x and as or (ii) is positive for all x and as . In either case, we have that as . This contradicts the fact that is bounded above by . (Get it? means that gets arbitrarily close to 1 for sufficiently large values of x; in particular, it can get within of 1, in which case, we would have the contradiction that for these sufficiently large values of x.)
Hence g must cross the horizontal axis exactly once; in other words, f has a unique fixed point.
For this, just choose any strictly increasing function such that (or such that ). For example, , i.e. .