Question

Let $f:[4,\infty [\to [4,\infty [$ be defined by $f\left(x\right)=5^{x(x-4)}$. Then $f^{-1}\left(x\right)$ 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. Not defined

Answer 1

The correct option is D Not defined

A function $f\left(x\right)$ has inverse function $f^{-1}\left(x\right)$ when $f\left(x\right)$ is bijective (one-one and onto) function
$f\left(x\right)=5^{x(x-4)}$
$f^{\prime }\left(x\right)=(2x-4)5^{x(x-4)}\log 5>0[4,\infty )$
$f\left(x\right)$ is one-one and also strictly increasing in $(4,\infty )$
$x=4\Rightarrow f\left(4\right)=5^0=1$
when $=\infty \Rightarrow f\left(\infty \right)=5^{\infty }=\infty$
$\therefore$ Range $=[1,\infty [$
$\therefore$ Range $\ne$ codomain
i.e., $[1,\infty [\ne [4,\infty [$
$f\left(x\right)$ is not onto
$f^{-1}\left(x\right)$ is not defined. Therefore $f^{-1}\left(x\right)$ does not exist.