# The function f is defined as f(x) = x/(x-1), how do you find f(f(x))?

Apr 8, 2017

Substitute f(x) for every x and then simplify.

#### Explanation:

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

Substitute f(x) for every x

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

Multiply numerator and denominator by 1 in the form of $\frac{x - 1}{x - 1}$

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

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

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

$f \left(f \left(x\right)\right) = x$

This means that $f \left(x\right) = \frac{x}{x - 1}$ is its own inverse.