Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
0 votes

Using the definition,prove that the function $f:\;A\rightarrow B$ is invertible if and only if f is both one-one and onto.

Can you answer this question?

1 Answer

0 votes
  • A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function
  • A function$ f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
  • A function f is invertible if and only if f is one-one and f is onto. also $f^{-1}$ is a function g such that $fog=I=gof$
Given:If f is invertible then f must be one -one and onto ie then exist $g=f^{-1}$ such that $ fog=I=gof$
Step1: Injective or One-One function:
Also $f^{-1}(f(y))=y$
f is one-one
Step 2: Surjective or On-to function:
and for every element y belonging h image f there exists $f(x)=y$
f is onto
If f is one one and onto by definition
f is invertible. ie then exists $g =f^{-1}$
Such that $gof=fog=I$



answered Mar 4, 2013 by meena.p
edited Mar 27, 2013 by meena.p

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App