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Mathematics –Class XII

Unit-I:Relations and functions

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Chapter-1: Relations and functions

Sub Topic: Inverse of function

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Outline

  • Definition
  • Working Rule
  • Examples
  • Assignment

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Bijective function

Any function which is one-one and

onto,is known as bijective function

One-one onto function

Bihar

Patna

Chandigarh

M.P.

Goa

Bhopal

Panji

Haryana

State

City

f=Capital

=> n(Domain)=n(co-domain)

This is bijective function

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Any function is

invertible only when

it is one-one onto

(Bijective)

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If any function is bijective,then

it is invertible and its inverse is f-1

Let f:A🡪B be one one onto function .Let y be any arbitrary element of B. Then f, being onto, there exists an element xϵA such that y=f(x) . Also f, being one one ,

this x must be unique.

Thus for each y ϵB ,there exists a unique xϵ A , s. t., y=f(x).

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So we may define a function ,denoted by f-1 as:

f-1 :B🡪A , such that f-1 (y)=x ⬄ f(x)=y

f-1 is called as inverse of f

Df-1 =Rf and Rf-1 =Df

>

f

A

B

>

f-1

If y=f(x), then

x=f-1(y)

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Working Rule to find inverse function

If f:A🡪B is defined as y=f(x)

Step 1.Prove that f is one-one i.e., take f(x1)=f(x2) and show that x1 = x2

Step 2.Prove that f is onto i.e., for any yϵB, there exists xϵA s.t. f(x) =y

Step 3. Find x in terms of y from y=f(x).

This is the inverse of function.

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Solution:- We have f(x)=x2+1.

Example:-Show that f:R🡪R given by

f(x)=x2+1 is not invertible

and domain is R. So 2 and -2 both are in domain

but f(-2)=(-2)2 +1=5 and f(2)=(2)2 +1=5

  • f(-2)=f(2) whereas -2≠2.

Hence f is not one-one function.

  • f is not invertible

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Example:-Let f:R-{3}🡪R-{1} be a function defined as .

Show that f is one-one onto.Hence find the inverse of f

Hence f is

one-one

Soln .Firstly we have to show that f is one-one

For this let f(x1 )=f(x2 )

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ONTO:-

let y be an arbitrary element of Co-domain then

From here it is clear that , ∀y except 1 there exists xϵDomain s. t. , y=f(x).

=> f is onto

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Thus f is one-one onto (bijective) function

Therefore invertible

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Q-1.Let f:R+ 🡪[-5, ∞[ given by

f(x)=9x2+6x-5.Show that f is invertible.

Also find the inverse of f.

Q.2-Let f:N🡪R be a function defined as

f(x)=4x2+12x+15.Show that f:N🡪Rf is

invertible. Find the inverse of f

Q.3-Let f:R-{-4/3}🡪R be a function defined as .

Show that f is one-one onto.

Hence find the inverse of f

Assignment