Let f : R → R be any function. Define g : R → R by g (x) = ¦f (x) ¦for all x.
Then g is :
Onto if f is onto
One-one is f one-one
Continuous if, f is continuous
Differentiable if f is differentiable

This can be simply explained through an example:

let f(x) = x  so g(x) = |x|

so if f(x) is continuous and differentiable , then g(x) is continuous and differentiable too

and if f(x) is one-one function , then g(x)cant be one-one but g(x) will be an onto fuction or many to one function

and f(x) is onto , then g(x) will be an onto function too

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regards

Ramesh

thats a shockingly atrocious reply by the expert

Its obvious that g(x) is non-negative. Hence for negative real values there is no x such that g(x) = y. So onto is ruled out.

Again if x and y are two real numbers such that f(x) = - f(y) we have g(x) = g(y) though x /= y . So one-one is out.

Continuous is right, as if a sequence {x} is convergent to l, {|x|} is convergent to |l|

Differentiability-the critical points are where f(x) = 0. If f(x) changes sign in an interval containing this point and if f'(x) /= 0 at this point then we have a point of non-differentiability.