Question

If $f:R\to R^{+}$ defined by $f\left(x\right)=e^{5x-1},$ then $f^{-1}\left(x\right)=$

• A. $\ln x+1$
• B. $\frac{1}{5}(\ln x+1)$
• C. $\ln 5x+1$
• D. $\frac{1}{5}(\ln 5x+1)$

Answer 1

The correct option is B $\frac{1}{5}(\ln x+1)$

It can be inferred from the options that $f$ is a bijective function.
$y=e^{5x-1}$
$\Rightarrow \ln y=\ln \left(e^{5x-1}\right)$
$\Rightarrow \ln y=5x-1$
$\Rightarrow \ln y+1=5x$
$\Rightarrow \frac{\ln y+1}{5}=x$
Swapping $x$ with $y$,
$y=\frac{1}{5}(\ln x+1)$
$\Rightarrow f^{-1}\left(x\right)=\frac{1}{5}(\ln x+1)$