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Question 9
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
Correct Answer: 12
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
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