Question

(a) Let = { ∈ : − 1 < < 1} and : → be defined by () =



1+||

, ∈ .

Show that is a bijection.

​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

A=R−{3} and B=R−{1}

one−one:

We have, f(x)=

x−3

x−2

Calculate f(x

1

):

⇒ f(x

1

)=

x

1

−3

x

1

−2

Calculate f(x

2

):

⇒ f(x

2

)=

x

2

−3

x

2

−2

Now, f(x

1

)=f(x

2

)

⇒

x

1

−3

x

1

−2

=

x

2

−3

x

2

−2

⇒ (x

1

−2)(x

2

−3)=(x

2

−2)(x

1

−3)

⇒ x

1

x

2

−3x

1

−2x

2

+6=x

1

x

2

−3x

2

−2x

1

+6

⇒ −x

1

=−x

2

⇒ x

1

=x

2

∴ f is an one-one function.

onto:

Let y∈B=R−{1}

Then, y



=1.

The function f is onto if there x∈A such that f(x)=y.

Now,

⇒ f(x)=y

⇒

x−3

x−2

=y

⇒ x−2=xy−3y

⇒ x(1−y)=−3y+2

⇒ x=

1−y

2−3y

∈A [ y



=1]

Thus, for any y∈B, there exists

1−y

2−3y

∈A such that

⇒ f(

1−y

2−3y

)=

(

1−y

2−3y

)−3

(

1−y

2−3y

)−2

=

2−3y−3+3y

2−3y−2+2y

=

−y

−y

=y

∴ f(

1−y

2−3y

)=y

∴ f is onto.