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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Using the definition,prove that the function $f:\;A\rightarrow B$ is invertible if and only if f is both one-one and onto.

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Toolbox:
  • 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:
 
$f^{-1}of(x)=x$
 
$f^{-1}(f(x))=x$
 
Also $f^{-1}(f(y))=y$
 
$=>f(x)=f(y)=>x=y$
 
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
 

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