Question

If $f\left(x\right)$ is an invertible function and $g\left(x\right)=2f\left(x\right)+5$, then the value of $g^{-1}\left(x\right)$, is

• A. $2f^{-1}\left(x\right)-5$
• B. $\frac{1}{2f^{-1}\left(x\right)+5}$
• C. $\frac{1}{2}{f}^{-1}\left(x\right)+5$
• D. $f^{-1}\left(\frac{x-5}{2}\right)$

Answer 1

The correct option is D $f^{-1}\left(\frac{x-5}{2}\right)$

$g\left(x\right)=2f\left(x\right)+5$
Replacing $x$ by $g^{-1}\left(x\right)$, we get
$x=2f\left(g^{-1}\right(x\left)\right)+5$
$\therefore f\left(g^{-1}\right(x\left)\right)=\frac{x-5}{2}$
$\therefore g^{-1}\left(x\right)=f^{-1}\left(\frac{x-5}{2}\right)$