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Let \mathrm{f: \mathbf{R} \rightarrow \mathbf{R}} be any function. Define \mathrm{g: \mathbf{R}\rightarrow \mathbf{R}\: by \: g(x)=|f(x)| for\; all \: x}.Then \mathrm{\mathrm{g}} is

Option: 1

g may be bounded even if f is unbounded


Option: 2

one-one if f is one


Option: 3

continuous if f is continuous


Option: 4

differentiable if f is differentiable

 


Answers (1)

best_answer

Take \mathrm{f(x)=x}. Since \mathrm{g}: \mathbf{R} \rightarrow \mathbf{R} given by \mathrm{g(x)=|x|} is not one-one so (b) is violated. Also g is not differentiable at \mathrm{x=0}. Let \mathrm{u(x)=|x|} then u is continuous function and \mathrm{\mathrm{g}(x)=u(f(x))=u\: of\: f(x)}.Hence \mathrm{\mathrm{g}} is continuous if \mathrm{\mathrm{f}} is continuous. It is easy to see \mathrm{\mathrm{g}} is bounded if and only if \mathrm{\mathrm{f}} is bounded.

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rishi.raj

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