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Even And Odd Function
A function f(x) : X → Y defined such that
f(-x) = f(x) ∀ x ε X
is called an even function and
if f (-x) = -d(x) ∀ x ε x, then the function f(x) is called an odd function.
Graphically, an even function is symmetrical w.r.t. y-axis and odd function is symmetrical w.r.t. origin.
Note : In general all functions can be represented as sum of an even function and an odd function.
Let, a function be defined as y = f(x). It can be written as:
=> y = (f(x) + f(-x))/2 + (f(x) - f(-x))/2
y = F1(x) + F2(x)
F1(-x) = (f(x) + f(-x))/2 = F1(x)
And F2(-x) = (f(-x) - f(x))/2
= -((f(x) - f(x))/2)
Here F1(x) is an even function and F2(x) is an odd function.
State whether the following functions are odd or even or neither.
(1) y = x3
(2) y = x4
(3) y + x + cos x
(4) y = loge(x + √(x2 + 1))
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