Question

If $f\left(x\right)=\frac{x+1}{x-1}$ show that f(f(x)) is an identify function.

• A. $1$
• B. $-1$
• C. $x$
• D. $-x$

Answer 1

The correct option is C $x$

$f\left(x\right)=\frac{x+1}{x-1}$

$f\left(f\left(x\right)\right)=\frac{f\left(x\right)+1}{f\left(x\right)-1}$

$=\frac{\left(\frac{x+1}{x-1}\right)+1}{\left(\frac{x+1}{x-1}\right)-1}$

$=\frac{\frac{x+1+x-1}{x-1}}{\frac{x+1-x+1}{x-1}}$

$=\frac{\frac{2x}{x-1}}{\frac{2}{x-1}}$

$=\frac{2x}{2}=x$

Now, $f\left(f\left(x\right)\right)=x$, which is an identity function.