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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∗?
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∗?
CBSE | Class 12 | Maths | NCERT | Relations and Functions
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∗?

SOLUTON:

Given: f : R∗ → R∗ characterized by f(x) = 1/x

Check for One-One

????(????1) = ???? ???????????? ????(????2) = ????1

???????? ????(????1) = ????(????2) ????ℎ???????? ????= x2

This infers ????1 = ????2

In this way, f is one-one capacity.

Check for onto

f(x) = 1/x or y = 1/x or x = 1/y f(1/y) = y

In this way, f is onto work. Once more, If f(x1) = f(x2)

Say, n1, n2 ∈ R

1/n1= 1/n2

So n1 = n2

In this way, f is one-one

Each genuine number having a place with co-space might not have a pre-picture in N. for instance, 1/3 and 3/2 are not has a place N. So N isn’t onto.

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