Relations and Functions-eleventh grade-9

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Show that every function f(x) can be represented as the sum of an even function and an odd function.

Solution: Let f(x) = f₁ (x) +f₂ (x)

Let f₁ (x) be an even function

Therefore, f₁ (-x) = f₁ (x)

Let f₂ be an odd function

Therefore, f₂ (-x) = - f₂

Now f(x) = f₁ (x) + f₂ (x) …………………………(i)

f(-x) = f₁ (-x) + f₂ (-x)

= f₁ (x) - f₂ (x) …………………………(ii)

Adding I and II, we get

f(x) + f(-x) = 2 f₁ (x)

Therefore, f₁ (x) = { f(x) + f(-x) }/2 …………………………(iii)

Subtracting II from I , we get

f(x) – f(-x) = 2 f₂ (x)

Implies, f₂ (x) = {f(x) – f(-x)}/2 …………………………(iv)

Therefore, f(x) = f₁ (x) + f₂ (x)

= { f(x) + f(-x) }/2 + {f(x) – f(-x)}/2 (from III and IV)