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Let f(x) = ax2 + bx + c where a, b, c are real constants. If f(x + y) = f(x) + f(y) for all real x and y then |
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Answer» \(f(x) = ax^2 + bx + c, \;a, b, c \in R\) \(f(x+ y) = f(x) + f(y)\) Take y = 0 \(f(x+ 0) = f(x) + f(0)\) ⇒ \(f(0) = f(x) - f(x) = 0\) ⇒ \(C = 0 \) \((\because f(0) = C)\) \(\therefore f(x) = ax^2 + bx \) Take y = x Then \(f(x + x) = f(x) + f(x)\) ⇒ \( f(2x) = 2f(x)\) ⇒ \(f(2x) = 2f(x) = 2(ax^2 + bx)\) ⇒ \(a(2x)^2 + b(2x) = 2(ax^2 + bx)\) ⇒ \(4ax^2 + 2bx = 2ax^2 + 2bx\) ⇒ \(2ax^2 = 0\) ⇒ \(a= 0\) (Because it is true for all x & y) \(\therefore f(x) = bx \) which is a linear function |
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