LimitsHard
Question
Let f: R → R be any function and g(x) = 
. Then g is
. Then g isOptions
A.onto if f is onto
B.one-one if f is one-one
C.continuous if f is continuous
D.differentiable if f is differentiable
Solution
Given g(x) = 
g(x) cannot be define where f(x) = 0
⇒ if f(x) is onto, g(x) may or may not be onto
same argument can be given for continuity and differentiability of g(x)
if f(x) is one-one
⇒ for x1 ≠ x2 ⇒ f(x1) ≠ f(x2)
⇒ g(x1) ≠ g(x2)
⇒ g(x) is also one-one
g(x) cannot be define where f(x) = 0
⇒ if f(x) is onto, g(x) may or may not be onto
same argument can be given for continuity and differentiability of g(x)
if f(x) is one-one
⇒ for x1 ≠ x2 ⇒ f(x1) ≠ f(x2)
⇒ g(x1) ≠ g(x2)
⇒ g(x) is also one-one
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