Question

F=(-1;1) range of f is given by f(x) =x\x+2is one one and onto find inverse also

Answer 1

Step-by-step explanation:

Let x

1

,x

2

∈D

f

f(x

1

)=f(x

2

)→

x

1

+2

x

1

=

x

2

+2

x

2

⇒x

1

(x

2

+2)=x

2

(x

1

+2)

⇒x

1

x

2

+2x

1

=x

1

x

2

+2x

2

⇒x

1

=x

2

hence f is one-one

let y=

x+2

x

⇒xy+2y=x

⇒xy−x=−2y

⇒x(1−y(∣)=2y

⇒x=

1−y

2y

∀ y ∈ R,∃ x ∈ D

f

such that

f(

1−y

2y

)=y, hence f is onto

∴ Inverse exist

∴f

−1

(x)=

1−x

2x

∀ x ∈ D

f−1

=R

f

=R−{1}