Given f(x)=1/(x-2), how do you find f(f(x))?

Mar 31, 2016

$f \left(f \left(x\right)\right) = \text{ } \frac{x - 2}{2 x + 5}$

Explanation:

Where ever you see $x$ replace it with $\frac{1}{x - 2}$

$f \left(f \left(x\right)\right) = \frac{1}{\left(\frac{1}{x - 2}\right) - 2}$

$\textcolor{b l u e}{\text{Consider just the denominator}}$

Multiply the 2 by $1$ but in the form of $1 - \frac{x - 2}{x - 2}$

So the denominator becomes

$\frac{1}{x - 2} - \left(\frac{2}{1} \times \frac{x - 2}{x - 2}\right)$

(1-2(x-2))/(x-2

$\frac{1 + 2 x + 4}{x - 2}$

$\frac{2 x + 5}{x - 2}$

$\textcolor{b l u e}{\text{Substitute into original expression}}$

$\textcolor{g r e e n}{f \left(f \left(x\right)\right) = 1 \div \frac{2 x + 5}{x - 2} \text{ "=" } \frac{x - 2}{2 x + 5}}$