Share

# 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

#### 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

#### Video TutorialsVIEW ALL [5]

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.
S