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, f(x)0, 1
For x=0, f(x)=0
For all x<0, f(x)-1, 0

So, clearly f(x) is not onto, as its range belongs to only -1, 1, and not (-R, R).

Now, f '(x)= 11+x2ddx(x) + xddx11+x2         =11+x2+x-2x1+x22         =11+x2 - 2x21+x22

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 x1 & x2 for which f(x1)=f(x2).
So, f(x) is not one-one.

Thus, it has been proved that f(x) is neither one-one nor onto.

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