Express a Function as the Sum of an Even and an Odd Functions

Tutorials on how to express any function in terms of an even and an odd function are presented with examples in the form of questions including detailed solutions and explanations.

Question 1:
Show that any function f may be expressed as the sum of an even and an odd functions.

Solution to Question 1:

Let us write f(x) as follows
f(x) = (1 / 2) f(x) + (1 / 2) f(x) + (1 / 2) f(-x) - (1 / 2) f(-x)
= (1 / 2) (f(x) + f(-x)) + (1 / 2)(f(x) - f(-x))
Let .
g(x) = (1 / 2) (f(x) + f(-x))
Check that g(x) is even
g(-x) = (1 / 2) (f(-x) + f(x)) = g(x)
Let .
h(x) = (1 / 2) (f(x) - f(-x))
Check that h(x) is odd
h(-x) = (1 / 2) (f(-x) - f(x)) = - (1 / 2) (f(x) - f(-x)) = - h(x)
Hence
f(x) = g(x) + h(x)
where g(x) is even and h(x) is odd and are defined in terms of f(x) above.


Question 2:
Express f(x) = 2x4 - 5 x3 + 2x2 + x - 4 as the sum of an even and an odd functions.

Solution to Question 2:

f(x) is a polynomial and it is therefore straightforward to separate even and odd parts of the polynomial as follows
f(x) = (2 x4 + 2 x2 - 4) + (- 5 x3 + x)
where 2 x 4 + 2 x 2 - 4 is a n even function and -5 x3 + x is an odd function.


Question 3:
Express f(x) = 1 / (x - 1) as the sum of an even and an odd functions and check your answer.

Solution to Question 3:

In exercise 1, we showed that any function f may be expressed as the sum of an even and an odd functions as follows
f(x) = (1 / 2) (f(x) + f(-x)) + (1 / 2)(f(x) - f(-x))
where g(x) = (1 / 2) (f(x) + f(-x)) is an even function and h(x) = (1 / 2)(f(x) - f(-x)) is an odd function. Hence if f(x) = 1 /(x - 1), then
g(x) = (1 / 2)(1 / (x - 1) + 1 / (- x - 1)) = 1 / (x2 - 1)
h(x) = (1 / 2)(1 / (x - 1) - 1 / (- x - 1)) = x / (x2 - 1)
We now check our answer. We add g(x) and h(x) and see if the addition gives f(x)
g(x) + h(x) = 1 / (x2 - 1) + x / (x2 - 1)
= ( 1 + x ) / (x2 - 1)
= (1 + x) / [(x - 1)(x + 1)]
= 1 / x - 1 = f(x)


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