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

Answer 1

Correct Answer: 12

Given: 

1. \( f(x) + f(x - 1) = 1 \)
2. \( f(x^2 - x) = 5 \)
3. \( g(x) = x^2 \)

Step 1: Substituting \( x = 1 \) in equations (1) and (2):

• \( f(0) = 5 \) from equation (2), since \( x^2 - x = 0 \)
• \( f(1) + f(0) = 1 \Rightarrow f(1) = 1 - 5 = -4 \)

Step 2: Substituting \( x = 2 \) in equation (1):

• \( f(2) + f(1) = 1 \Rightarrow f(2) = 1 - (-4) = 5 \)

Observation:

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

Final Calculation:

• \( f(g(5)) + g(f(5)) = f(25) + g(-4) \)
• \( f(25) = -4 \) since 25 is odd
• \( g(-4) = (-4)^2 = 16 \)
• \( f(g(5)) + g(f(5)) = -4 + 16 = 12 \)