CAT 2022 Slot 2 QA Question 10: 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 $f(x^2-x)=5$, then the value of the sum $f(g(5))+g(f(5))$ is

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 $f(x^2-x)=5$, then the value of the sum $f(g(5))+g(f(5))$ is

Accepted Answer

From the given condition:

$f(x) + f(x - 1) = 1$

This gives $f(x) = 1 - f(x - 1)$. Applying it again:

$f(x - 1) + f(x - 2) = 1$
$f(x - 2) = 1 - f(x - 1) = f(x)$

So, $f$ has period 2 over integer inputs.

Also, $x^2 - x = x(x - 1)$, which is always even for integer $x$. Since $f(x^2 - x) = 5$, $f(

Suppose for all integers xxx, there are two functions fff and ggg such that f(x)+f(x−1)−1=0f(x)+f(x-1)-1=0f(x)+f(x−1)−1=0 and g(x)=x2g(x)=x^2g(x)=x2. If f(x2−x)=5f(x^2-x)=5f(x2−x)=5, then the value of the sum f(g(5))+g(f(5))f(g(5))+g(f(5))f(g(5))+g(f(5)) is

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 $f(x^2-x)=5$ , then the value of the sum $f(g(5))+g(f(5))$ is