Question 1037247
Let {{{f(x) = x^2 - x + 3}}}.

f has no inverse function since it fails the horizontal line test for one-to-one-ness.
f(x) > x since, for all real x, {{{x^2-2x+3 >0}}}, hence {{{f(x) = x^2 - x + 3 > x}}}.
Finally f(1) = 3 > 0.

{{{graph( 300, 200, -5, 5, -5, 5, x^2 - x +3, x )}}}.

(Notice that the parabola is entirely above the line y = x, which signifies f(x) > x for all x.)

There still many more functions that would satisfy the initial conditions.