How to solve the equation $(f \circ f)(x)=x$?

Question

Solve the equation $(f \circ f)(x)=x$, if $f(x) = \frac {2x+1}{x+2}$ and $x \in \mathbb R$ \ $\{-2\}$.

How would I solve this equation and what does it even mean to be solved in this context?

Answer 1

If $(f\circ f)(x)=x$, then $$\frac{2f(x)+1}{f(x)+2}=x$$ $$\frac{2\frac{2x+1}{x+2}+1}{\frac{2x+1}{x+2}+2}=x$$ And after you solve it (it should turn into a quadratic), don't forget to cut out the extraneous solutions!

Answer 2

Solve $\dfrac{2\left(\dfrac{2x+1}{x+2}\right)+1}{\dfrac{2x+1}{x+2}+2}=x$ for $x\ne-2$.

Answer 3

How to begin: $$f\bigl(f(x)\bigr)=\frac{2f(x)+1}{f(x)+2}=\frac{2\cfrac{2x+1}{x+2}+1}{\cfrac{2x+1}{x+2}+2}=\cdots$$