Question

Let f : R → R be a function. Define g:R → by g(x) = |f(x)| for all x. Then g is

A
Onto if f is onto
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B
One-one if f is one-one
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C
Continuous if f is continuous
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D
None of these
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Solution

The correct option is C Continuous if f is continuousg(x)=|f(x)|≥0.So g(x) cannot be onto. if f(x) one-one and f(x1)=−f(x2) then g(x1)=g(x2). So, 'f(x) is one-one' does not ensure that g(x) is one-one. If f(x) is continuous for x∈R, (x) is also continuous for x∈R. This is obvious from the following graphical consideration. So g is continuous if f is continuous.

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