CAT 2022 Slot 3 QA Q6 Functions & Graphs

Question 6 of the Quantitative Ability (QA) section, CAT 2022 Slot 3, with CAT's official answer and a worked solution.

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

A
\(x \leq r\)
B
\(x \geq r\)
C
\(x \gt r\)
D
\(x \neq r\)

Answer: (A) x ≤ r Functions & Graphs MCQ

Solution

If \(x\lt r\): \(f(x)=r\) and \(f(r)=2r-r=r\), so \(f(f(x))=f(x)\). Holds.
If \(x\ge r\): \(f(x)=2x-r\ge r\), so \(f(f(x))=2(2x-r)-r=4x-3r\).
\(4x-3r=2x-r\) only when \(x=r\). So the equation holds exactly for \(x\le r\).