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1. relation and function — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths1. relation and function1 Mark
MCQ
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} $
  • ✓
    $2 + \sqrt {4 + {{\log }_5}\ x} $
  • C
    ${\left( {\frac{1}{5}} \right)^{x\left( {x - 4} \right)}}$
  • D
    $2 + \sqrt {4 - {{\log }_5}\ x} $
✓

Answer

Correct option: B.
$2 + \sqrt {4 + {{\log }_5}\ x} $
b
$ 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} $

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