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1.Relation and Function
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Let $f:\left[ {4,\infty } \right) \to \left[ {1,\infty } \right)$ be a function defined by $f\left( x \right) = {5^{x\left( {x - 4} \right)}}$ then $f^{-1}(x)$ is
A
$2 - \sqrt {4 + {{\log }_5}\ x} $
B
$2 + \sqrt {4 + {{\log }_5}\ x} $
C
${\left( {\frac{1}{5}} \right)^{x\left( {x - 4} \right)}}$
D
$2 + \sqrt {4 - {{\log }_5}\ x} $
Solution
$ 5^{x(x-4)}=y $
$ \Rightarrow x^{2}-4 x=\log _{5} y $
$\Rightarrow (x-2)^{2}=\log _{5} y+4 $
$ \Rightarrow x=2 \pm \sqrt{\log _{5} y+4} $
$ \Rightarrow x=2+\sqrt{\log _{5} y+4} \quad(\because x \in[4, \infty)) $
$\Rightarrow \quad f^{-1}(x)=2+\sqrt{\log _{5} x+4} $
Standard 12
Mathematics
