3. (a) Determine whether f(x)is even or odd or neither(b) Determine whether g(x) = 5 - 4z4 +323 - 2x2 + x _ 1 is even or odd or neither(c) Show that any function can be written as the sum of an even and an odd functionShow how g(z) above can be written as the sum of an even and an odd function

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Asked Nov 18, 2019
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Practice Question 3.3

3. (a) Determine whether f(x)
is even or odd or neither
(b) Determine whether g(x) = 5 - 4z4 +323 - 2x2 + x _ 1 is even or odd or neither
(c) Show that any function can be written as the sum of an even and an odd function
Show how g(z) above can be written as the sum of an even and an odd function
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3. (a) Determine whether f(x) is even or odd or neither (b) Determine whether g(x) = 5 - 4z4 +323 - 2x2 + x _ 1 is even or odd or neither (c) Show that any function can be written as the sum of an even and an odd function Show how g(z) above can be written as the sum of an even and an odd function

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Expert Answer

Step 1

Let g(x) be any function we can write it as shown below

1
8 (x)())(-x)(-)
1
1
(2 (x)+(-x)(2(x)-g(-x)
t()()(-) ) md k()=(()-2(-3)
8 (x) f(x)+h(x)
1
(g(x)+g(-x))and h(x)=(g(x)-(-x)
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1 8 (x)())(-x)(-) 1 1 (2 (x)+(-x)(2(x)-g(-x) t()()(-) ) md k()=(()-2(-3) 8 (x) f(x)+h(x) 1 (g(x)+g(-x))and h(x)=(g(x)-(-x)

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Step 2

Here f(x) and h(x) are even and o...

s(x)(()+g(-x))
Substitute x in f (x ) we get
f(-x)=(8(-x)8(+x))
gg
f(-x)=(x)
Thus f(x)is an even function
Mx)(s(x)-(- ))
Substitute x in h (x) we get
h(-x)(x)-(+x))
M(-) )-g(-))
h(-x)h(x)
Thus g(x)is an odd function
Thus, we can write any function g(x) as the sum of even and odd functions
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s(x)(()+g(-x)) Substitute x in f (x ) we get f(-x)=(8(-x)8(+x)) gg f(-x)=(x) Thus f(x)is an even function Mx)(s(x)-(- )) Substitute x in h (x) we get h(-x)(x)-(+x)) M(-) )-g(-)) h(-x)h(x) Thus g(x)is an odd function Thus, we can write any function g(x) as the sum of even and odd functions

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