Question

Let $a>1$ be a real number and $f\left(x\right)={\log }_a{x}^2$ for $x>0$. If $f^{-1}$ is the inverse function of $f$ and $b$ and $c$ are real numbers then $f^{-1}(b+c)$ is equal to

• A. $f^{-1}\left(b\right).f^{-1}\left(c\right)$
• B. $f^{-1}\left(b\right)+f^{-1}\left(c\right)$
• C. $\frac{1}{f(b+c)}$
• D. $\frac{1}{f^{-1}\left(b\right)+f^{-1}\left(c\right)}$

Answer 1

The correct option is A $f^{-1}\left(b\right).f^{-1}\left(c\right)$

$y={\log }_a\left(x^2\right)$
$a^y=x^2$
$a^{y/2}=x$
Hence $f^{-1}=a^{x/2}$
$f^{-1}(b+c)$
$=a^{(b+c)/2}$
$=a^{b/2}{a}^{c/2}$
$=f^{-1}\left(b\right)f^{-1}\left(c\right)$
Hence, option 'A' is correct.