MHT CET202615 April 2026Morning ShiftMathematicsFunctionsActual
If f(x-y) + f(x+y) = 2f(x)f(y) for all x, y R , then f(x) is .....
Options
- Aan odd function
- Ban even function
- Cneither even nor odd function
- Da periodic function
Correct answer
B. an even function
Step-by-step solution
Given the functional equation: f(x-y) + f(x+y) = 2f(x)f(y) Substituting y = 0 , we get: f(x) + f(x) = 2f(x)f(0) 2f(x) = 2f(x)f(0) f(x)(1 - f(0)) = 0 This implies that either f(x) = 0 for all x R , or f(0) = 1 . Case 1: If f(x) = 0 for all x , then f(-x) = 0 = f(x) . Thus, f(x) is an even function. Case 2: If f(0) = 1 , substitute x = 0 in the given functional equation: f(0-y) + f(0+y) = 2f(0)f(y) f(-y) + f(y) = 2(1)f(y) f(-y) = f(y) This shows that f(x) is an even function. In all possible cases, f(x) is an even fu