Answer
The value of x which satisfies the equation $8+{{f}^{-1}}\left( x-1 \right)=10$ is $x=7$.
Work Step by Step
$8+{{f}^{-1}}\left( x-1 \right)=10$
Solve the equation further,
$\begin{align}
& {{f}^{-1}}\left( x-1 \right)=10-8 \\
& {{f}^{-1}}\left( x-1 \right)=2 \\
\end{align}$ (I)
Also, $f\left( 2 \right)=6$
Take the inverse of the above function ${{f}^{-1}}\left( f\left( 2 \right) \right)={{f}^{-1}}\left( 6 \right)$
As ${{f}^{-1}}\left( f\left( x \right) \right)=x$, therefore${{f}^{-1}}\left( f\left( 2 \right) \right)=2$
And, $2={{f}^{-1}}\left( 6 \right)$
From (1),
${{f}^{-1}}\left( x-1 \right)=2$
And,
$2={{f}^{-1}}\left( 6 \right)$
So,
${{f}^{-1}}\left( x-1 \right)={{f}^{-1}}\left( 6 \right)$
Now compare both sides,
$\begin{align}
& x-1=6 \\
& x=1+6 \\
& x=7
\end{align}$
Thus, the value of x = 7 satisfies $8+{{f}^{-1}}\left( x-1 \right)=10$.
