Question

Let $f :[4, \infty) \rightarrow[1, \infty)$ be a function defined by $f ( x )=5^{ x ( x -4)}$, then $f ^{-1}( x )$ is

• A. $2-\sqrt{4+\log _{5} x }$
• B. $2+\sqrt{4+\log _{5} x }$
• C. $\left(\frac{1}{5}\right)^{ x ( x -4)}$
• D. None of these

Answer 1

Answer: (B)

Let
$y=5^{x(x-4)} $
$\Rightarrow x(x-4)=\log _{5} y$
$ \Rightarrow x^{2}-4 x-\log _{5} y=0$
$\Rightarrow x=\frac{4 \pm \sqrt{16+4 \log _{5} y}}{2}$
$=\left(2 \pm \sqrt{4+\log _{5} y}\right)$
But $x \geq 4$,
so $x=\left(2+\sqrt{4+\log _{5} y}\right)$
$\therefore f ^{-1}( x )=2+\sqrt{4+\log _{5} x }$