Deriving functions Problem

Question

Suppose that g is a real valued, differentiable function whose derivative g' satisfies the inequality |g'(x)|less than or equal to M for all x in R.
Show that if epsilon is greater than 0 is small enough, then the real valued function f defined by f(x)=x+epsilon*g(x) is one to one and onto.
Recall that a function f is said to be "one to one" if x sub 1 does not equal x sub 2 implies that f(x sub 1) does not equal f(x sub 2), and f is said to be "onto" if for every real number y, there is a real number x such that f(x) = y.

Answer 1

The function f will be one to one onto if it is monotone increasing on R.  f will be monotone increasing if its slope is everywhere greater than zero.  All that is required for the slope to be everywhere greater than zero is for epsilon to be less than 1/M.