Q. 13.9( 118 Votes )
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It is given that f : R* → R* defined by
check for one-one:
For a function to be one-one, if f(x) = f(y) then x = y.
f(x) = f(y)
⇒ x = y
⇒Therefore, f is one – one.
We can see that y ϵ R, there exists , such that
⇒ f is onto.
Therefore, function f is one-one and onto.
Now, Let us consider g: N → R* defined by
Then, we get,
g(x1) = g(x2)
⇒ x1 = x2
⇒ g is one–one.
It can be observed that g is not onto as for 1.2 ϵ R there does not exist any x in N such that
Therefore, function g is one –one but not onto.
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