CAT 2022 Slot 2 QA Q10 Functions & Graphs

Question 10 of the Quantitative Ability (QA) section, CAT 2022 Slot 2, with CAT's official answer and a worked solution.

Suppose for all integers \(x\), there are two functions \(f\) and \(g\) such that \(f(x)+f(x-1)-1=0\) and \(g(x)=x^2\). If \(f\left(x^2-x\right)=5\), then the value of the sum \(f(g(5))+g(f(5))\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 12 Functions & Graphs TITA

Solution

\(f(x)+f(x-1)=1\), so \(f\) alternates: \(f(k)=5\) forces \(f(k\pm1)=-4\).
\(x^2-x\) is always even, so \(f(\text{even})=5\) and \(f(\text{odd})=-4\).
\(f(g(5))=f(25)=-4\) and \(g(f(5))=(-4)^2=16\). Sum \(=12\).