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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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If $f(x)=(4-(x-7)^3\},then\;f^{-1}(x)$=____________.

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Toolbox:
  • we define $f^{-1}(x)$ as a function .
  • such that $(fof^{-1})(x)=x$ ie $g=f^{-1}$ is the inverse function of f if $fog=gof =I$
$f(x)=4-(x-7)^3$
 
Let $y=4-(x-7)^3$
 
$(x-7)^3=4-y$
 
$x=(4-y)^{1/3}+7$
 
we define a function g(y) such that $g(y)=(4-y)^{1/3}+7$
 
we see that $(fog)(y)=f((4-y)^{1/3}+7)$
 
$=4-((4-y)^{1/3}+7-7)$
 
$=4-((4-y)^{1/3})^3$
 
$=4-(4-y)$
 
$=y$
 
Hence y is inverse of f
 
and $g(x)=f^{-1}(x)=(4-x)^{1/3}+7$

 

answered Mar 6, 2013 by meena.p
 

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