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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 - CBSE (Commerce) Class 12 - Mathematics

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Question

Show that the function fR* → 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

It is given that fR* → R* is defined by `f(x) = 1/x`

One-one:

f(x) = f(y)

`=> 1/x = 1/y`

=> x = y

f is one-one.

Onto:

It is clear that for y R*, there exists  ` x= 1/y in R ("Exists as y" != 0)`such that

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

f is onto.

Thus, the given function (f) is one-one and onto.

Now, consider function g: N → R*defined by

`g(x) = 1/x`

We have,

`g(x_1) = g(x_2) =>  1/x_1 = 1/x_2 => x_1 = x_2`

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) = `1/1.2`

Hence, function g is one-one but not onto.

  Is there an error in this question or solution?

APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 1: Relations and Functions
Q: 1 | Page no. 10
Solution 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 Concept: Types of Functions.
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