CAT 2022 Slot 2 QA Question & Solution

AlgebraMedium

Question

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

Solution

Given,

$f\left(x\right)+f\left(x-1\right)=1$ ...... (1)

$f\left(x^2-x\right)=5$ ...... (2)

$g\left(x\right)=x^2$

Substituting x = 1 in (1) and (2), we get

f(0) = 5

f(1) + f(0) = 1

f(1) = 1 - 5 = -4

f(2) + f(1) = 1

f(2) = 1 + 4 = 5

f(n) = 5 if n is even and f(n) = -4 if n is odd

f(g(5)) + g(f(5)) = f(25) + g(-4) = -4 + 16 = 12