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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Let A=[-1,1].Then,dicuss whether the following functions defined on A are one-one,onto or bijective$(iii)\quad h(x)\;=\;x|x| $

Note: This is the 3rd part of a  4 part question, which is split as 4 separate questions here.

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Toolbox:
  • 1.A function f on set A is one-one if $f(x)=f(y) =>x,y \in A$
  • 2.A function f on set A is onto if for every $ y \in A$ then exists $x \in A$ such that $f(x)=y$
  • 3.A function is bijective if it is both one-one and onto
(iii)$h(x)=x |x| \qquad x \in [-1,1]$
 
case I when x takes -ve values
 
Let $x_1=-p_1\qquad x_2=-p_2\qquad p_1p_2 +ve$
 
Let $ h(x_1)=h(x_2)$
 
$h(x_1)=-p_1|p_1|=h(x_2)=-p_2|p_2|$
 
$-p_1 \times p_1=-p_2p_2$
 
$p_1^2=p_2^2$
 
$=>p_1=p_2 \qquad (p_1,p_2 +ve)$
 
Let $h(x_1)=h(x_2)$
 
case I when x takes +ve values
 
$x_1=q_1 \qquad x_2=q_2$
 
$h(x_1)=q_1|q_1|=h(x_2)=q_2|q_2|$
 
$q_1^2=q_2^2$
 
$q_1=q_2$
 
Hence h is a one one function
 
Let $y \in [-1,1]$
 
$y=x|x|$
 
$=-x^2$ if y is -ve
 
$=x^2$ when y is +ve
 
and since $(-y)=x^2\; for\;y -ve$
 
$ x =\sqrt {-y} \qquad \in [-1,1]\; since\; y \;is\; -ve [-1,1]$
 
and $y=x^2$
 
$x=\sqrt y$ when y is +ve [-1,1]
 
Hence for every $ y \in [-1,1]$ then exists an element x in [-1,1] such that $R(x)=y$
 
R is onto function
 
Hence R is both one-one and onto

 

 

answered Mar 4, 2013 by meena.p
 

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