Question

Show that the function f: R* → R* defined byis 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*?

Answer 1

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*?

solution :- f:R* → R* defined by f(x) = 1/x
f(x) = f(y)
=> 1/x = 1/y
=> x = y
therefore , function f is one - one.
We can see that y ϵ R, there exists ,
x = 1/y ϵ R such that
$f(x)=\frac{1}{\frac{1}{y}}=y$
therefore, function f is onto.
Therefore, function f is one-one and onto.
Now, Let us consider g: N → R* defined by
g(x) = 1/x
Then, we get,
g(x₁) = g(x₂)
⇒1/x₁ = 1/x₂
⇒ x₁ = x₂
therefore, function g is one–one.
It can be observed that g is not onto as for 1.2ϵ R there does not exist any x in N such that
g(x) = 1/1.2
Therefore, function g is one –one but not onto