Given the function
[tex]f(x)=\sqrt[]{\frac{7^{x+6}}{7}}[/tex]
We are asked to find the inverse of the function.
Explanation
We will replace f(x) with y and interchange y with x
[tex]\begin{gathered} x=\sqrt[]{\frac{7^{y+6}}{7}} \\ \text{square both sides} \\ x^2=\frac{7^{y+6}}{7} \\ \text{cross multiply} \\ 7x^2=7^{y+6} \\ \\ \end{gathered}[/tex]
The next step is to introduce the log of 7 to both sides
[tex]\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}[/tex]
Answer:
[tex]\therefore f^{-1}(x)=\log _7(7x^2)-6[/tex]
