Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R* ?
It is given that f: R* → R* is defined by
One-one:
∴f is one-one.
Onto: It is clear that for y∈R*, there exists such that
∴f is onto.
Thus, the given function (f) is one-one and onto.
Now, consider function g: N → R* defined by
We have,
∴g is one-one.
Further, it is clear that g is not onto as for 1.2 ∈R* there does not exit any x in N such that g(x) =.
Hence, function g is one-one but not onto.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
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(ii) f : R → R defined by f(x) = 1 + x2
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y = Ax : xy' = y (x ≠ 0)
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