Problem 2-6: Odd and Even Functions

Which one of the following functions are even or odd or neither?
a) $\sin(x)$
b) $\cos(x)$
c) $\sin(x)\cos(x)$
d) $x \sin(x)$
e) $x^2$
f) $\sin(x) + \cos(x)$

Recall that a function is said to be even if $f(-x)=f(x)$ and odd if $f(-x)=-f(x)$.

Solution
a) Odd $\sin(-x)=-\sin(x)$

b) Even $\cos(-x)=\cos(x)$

c) Odd $\sin(-x)\cos(-x)=-\sin(x)\cos(x)$

d) Even $(-x) \sin(-x)=(-x) (- \sin(x))=x \sin(x)$

e) Even $(-x)^2=x^2$

f) Neither $\sin(-x) + \cos(-x)= -\sin(x) + \cos(x)$

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1 Comment

1. Abdulaziz says:

Please can you solve this example
X(t)=u(t-1)-u(t-3)
Express the signal using its even function xe(t) and Odd function xo(t)
Then sketch the original signal