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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Let f:$R\rightarrow R$ be the functions defined by $f(x)=x^3+5.$ Then $f^{-1}(x)$ is

\begin{array}{1 1}(A)\;(x+5)^{\frac{1}{3}} & (B)\;(x-5)^{\frac{1}{3}}\\(C)\;(5-x)^{\frac{1}{3}} & (D)\;5-x\end{array}

Can you answer this question?
 
 

1 Answer

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Toolbox:
  • A function g is inverse of $f:R \to R $ if $fog=gof=I_R\;ie \;g=f^{-1}$
$f:R \to R$
 
$f(x)=x^3+5$
 
Let $y=x^3+5 \qquad y \to R$
 
$x^3=y-5$
 
$x=(y-5)^{1/3}$
 
we define a function $g(y)=(y-5)^{1/3}$
 
$(gof)(x)=g(f(x))$
 
$=g(x^3+5)$
 
$=(x^3+5-5)^{1/3}=(x^3)^{1/3}=x$
 
Also $fog(g)=f(g(y))$
 
$=f\bigg((y-5)^{1/5}\bigg)$
 
$=\bigg[(y-5)^{1/5}\bigg]^3+5$
 
$=y-5+5$
 
=y
 
=>$fog=gof=I_R$
 
$f^{-1}(x)=g(x)=(x-5)^{1/3}$
 
'B' option is correct

 

 

answered Mar 5, 2013 by meena.p
 

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