Question

Find the inverse }^{3x - 1} " alt="f(x)={e}^{3x - 1} " align="absmiddle" class="latex-formula">
Tank you for your support ​

Answer 1

Answer:

$f^{-1}(x)=\dfrac{1}{3}(\ln x +1)$

Step-by-step explanation:

Given function:

$f(x)=e^{3x-1}$

Replace f(x) with y:

$\implies y=e^{3x-1}$

Take natural logs of both sides:

$\implies \ln y = \ln e^{3x-1}$

Apply the Power Log Law   $\ln x^n=n\ln x$ :

$\implies \ln y = (3x-1)\ln e$

As $\ln e=1$ then:

$\implies \ln y = 3x-1$

Rearrange to make x the subject:

$\implies 3x=\ln y +1$

$\implies x=\dfrac{1}{3}(\ln y +1)$

Swap x for $f^{-1}(x)$  and y for x:

$\implies f^{-1}(x)=\dfrac{1}{3}(\ln x +1)$