Let `f:R to R` be any function. Defining `g: R to R` by `g(x)=|f(x)|" for "x to R`. Then g, isA. onto if if is ontoB. one-one if f is one-oneC. continuous if f is continuousD. differentiable if f is differentiable

Let `f:R to R` be any function. Defining `g: R to R` by `g(x)=|f(x)|"

1.	Let `f:R to R` be any function. Defining `g: R to R` by `g(x)=\|f(x)\|" for "x to R`. Then g, isA. onto if if is ontoB. one-one if f is one-oneC. continuous if f is continuousD. differentiable if f is differentiable
Answer» Correct Answer - C Let `h(x)=\|x\|. "Then " h:R to R` is continuous many-one and into function. We have `hof (x)=h(f(x))=\|f(x)\|=g(x)` Since composition of continuous functions is continuous. Therefore, g(x) is continuous if f is continuous Since, composition of two bijections is a bijection. Here h(x) is many-one. So, g(x) cannot be one-one even if f is one-one. Also, g(x) cannot be onto even if f is onto. We observe that (f)=sin x is everywhere differentiable but \|sin x\|is not differentiable at `x=n pi, n in Z`. Therefore g(x) need not be differentiable even if f is differentiable