Question

Hi, how come f^-1(x)=2log7(x)-5 isn't listed as a choice for the response? Is it incorrect?

Answer 1

Given the function

`f(x)=\sqrt[]{(7^(x+6))/(7)}`

We are asked to find the inverse of the function.

Explanation

We will replace f(x) with y and interchange y with x

`\begin{gathered} x=\sqrt[]{(7^(y+6))/(7)} \\ \text{square both sides} \\ x^2=(7^(y+6))/(7) \\ \text{cross multiply} \\ 7x^2=7^(y+6) \\ \\ \end{gathered}`

The next step is to introduce the log of 7 to both sides

`\begin{gathered} \log _77x^2=\log _77^(y+6) \\ \log _77x^2=(y+6)\log _77 \\ \log _7(7x^2)=y+6 \\ y=\log _7(7x^2)-6 \\ \therefore f^(-1)(x)=\log _7(7x^2)-6 \end{gathered}`

Answer:

`\therefore f^(-1)(x)=\log _7(7x^2)-6`