Answer
The inverse of the function $f\left( x \right)$ is ${{f}^{^{_{-1}}}}\left( x \right)=\underline{\frac{10\left( x+1 \right)}{1-x}}$
Work Step by Step
We know that the inverse of a function $f\left( x \right)$ is labeled as ${{f}^{-1}}\left( x \right)$ and it satisfies:
$\left( f\circ {{f}^{-1}} \right)\left( x \right)=\left( {{f}^{-1}}\circ f \right)\left( x \right)$
For the inverse of an inverse of any function, find out the expression for x in terms of y.
Now, one can solve the function as shown below:
$\begin{align}
& f\left( x \right)=y \\
& \frac{x-10}{x+10}=y \\
& x-10=y\left( x+10 \right) \\
& x-10=yx+10y
\end{align}$
And solve for x as follows:
$\begin{align}
& x-10=yx+10y \\
& x-yx=10y+10 \\
& x\left( 1-y \right)=10\left( y+1 \right) \\
& x=\frac{10\left( y+1 \right)}{1-y}
\end{align}$
Now, replace all the y terms with x terms in the obtained function to get,
${{f}^{-1}}\left( x \right)=\frac{10\left( x+1 \right)}{1-x}$
Thus, the inverse of the function is $\frac{10\left( x+1 \right)}{1-x}$.
