# CAT Practice : Functions

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Keep finding f(x+1), f(x+2), f(x+3) and so on until one of them is equal to -f(x).

## Functions - Value of a

Q.10: Given that x is real and f(x) = f(x + 1) + f(x – 1). Determine the value of ‘a’ that will satisfy f(x) + f(x + a) = 0?
1. -1
2. -2
3. 1
4. 3

Choice D. 3

## Detailed Solution

f(x) = f(x + 1) + f(x – 1)
Let f(x) = p and f(x – 1) = q

f(x + 1) = f(x + 1 + 1) + f(x + 1 – 1) (Put x = x + 1 here!)
= f(x + 2) + f(x)
p – q = f(x + 2) + p
Or f(x + 2) = -q

f(x + 2) = f(x + 2 + 1) + f(x + 2 – 1) (Put x = x + 2 here!)
= f(x + 3) + f(x + 1)
– q = f(x + 3) + p – q
Or f(x + 3) = -p

At this point we notice that f(x) = p and f(x+3) = -p
f(x) + f(x + 3) = 0 (This is the condition to be satisfied to determine a)
Hence ‘a’ = 3

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Functions.