CAT 2022 Slot 3 QA Question 6: Let r be a real number and $f(x)=\begin{cases}2x-r, & ext{if } x \geq r\ r, & ext{if } x

Question

Let r be a real number and $f(x)=\begin{cases}2x-r, & 	ext{if } x \geq r\ r, & 	ext{if } x < r\end{cases}$. Then, the equation $f(x)=f(f(x))$ holds for all real values of x where

Accepted Answer

Consider the two cases in the definition of $f$.
If $x 
$f(f(x)) = f(r) = 2r - r = r$
So the equation holds for all $x 
If $x \geq r$, then $f(x) = 2x - r$. Also, $2x - r \geq r$, so the first branch applies again.
$f(f(x)) = f(2x - r)$
$= 2(2x - r) - r$
$= 4x - 3r$
For $f(x) = f(f(x))$ in this case

Let r be a real number and f(x)={2x−r,if x≥rr,if x<rf(x)=\begin{cases}2x-r, & \text{if } x \geq r\\ r, & \text{if } x < r\end{cases}f(x)={2x−r,r,​if x≥rif x<r​. Then, the equation f(x)=f(f(x))f(x)=f(f(x))f(x)=f(f(x)) holds for all real values of x where

Let r be a real number and $f(x)=\begin{cases}2x-r, & \text{if } x \geq r\\ r, & \text{if } x < r\end{cases}$ . Then, the equation $f(x)=f(f(x))$ holds for all real values of x where