Answer
$ f(\frac{1}{x-6})= \frac{x-5}{x-7}$.
Work Step by Step
The given function is
$\Rightarrow f(x)=\frac{1+x}{1-x}$
Replace $x$ with $\frac{1}{x-6}$.
$\Rightarrow f(\frac{1}{x-6})=\frac{1+\frac{1}{x-6}}{1-\frac{1}{x-6}}$
Multiply the numerator and the denominator by $(x-6)$.
$\Rightarrow f(\frac{1}{x-6})= \frac{(x-6)\left (1+\frac{1}{x-6}\right )}{(x-6)\left (1-\frac{1}{x-6}\right )}$
Use the distributive property.
$\Rightarrow f(\frac{1}{x-6})= \frac{1(x-6)+\frac{1(x-6)}{x-6}}{1(x-6)-\frac{1(x-6)}{x-6}}$
Simplify.
$\Rightarrow f(\frac{1}{x-6})= \frac{x-6+1}{x-6-1}$
$\Rightarrow f(\frac{1}{x-6})= \frac{x-5}{x-7}$.
