Show that the function f : R -- R defined by
f(x) =x / x2 +1 , ( x belongs R)
is neither one-one nor onto....
By a little examination, we can see that:-
For all x>0,
For x=0, f(x)=0
For all x<0,
So, clearly f(x) is not onto, as its range belongs to only , and not (-R, R).
So, f '(x)=0 at x=1.
This means, that the slope of f(x) changes sign before and after x=1.
That is, there exists some for which .
So, f(x) is not one-one.
Thus, it has been proved that f(x) is neither one-one nor onto.