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: Evaluate \( f(0) \) and \( f(1) \).

• From equation (2), \( f(x^2 - x) = 5 \). Setting \( x^2 - x = 0 \) yields \( x(x-1) = 0 \), so \( x = 0 \) or \( x = 1 \). Thus, \( f(0) = 5 \).
• From equation (1), substitute \( x = 1 \): \( f(1) + f(0) = 1 \). Substituting \( f(0) = 5 \), we get \( f(1) + 5 = 1 \), so \( f(1) = 1 - 5 = -4 \).

Step 2: Evaluate \( f(2) \).

• From equation (1), substitute \( x = 2 \): \( f(2) + f(1) = 1 \). Substituting \( f(1) = -4 \), we get \( f(2) + (-4) = 1 \), so \( f(2) = 1 + 4 = 5 \).

Pattern Identification:

• Based on the computed values, it appears that \( f(n) = 5 \) for even \( n \) and \( f(n) = -4 \) for odd \( n \).

Final Computation:

• We need to find \( f(g(5)) + g(f(5)) \).
• First, calculate \( g(5) = 5^2 = 25 \).
• Then, \( f(g(5)) = f(25) \). Since 25 is odd, \( f(25) = -4 \).
• Next, calculate \( f(5) \). Since 5 is odd, \( f(5) = -4 \).
• Then, \( g(f(5)) = g(-4) = (-4)^2 = 16 \).
• Finally, \( f(g(5)) + g(f(5)) = -4 + 16 = 12 \).