Is the function #2 sin x cos x# even, odd or neither?

1 Answer
Aug 21, 2015

Answer:

#f(x) = 2 sin x cos x# is an odd function.

Explanation:

#f(x) = 2 sin x cos x#

For any #x# in the domain of #f# (the domain is #RR#), #-x# is also in the domain of #f#, and:

#f(-x) = 2sin(-x)cos(-x)#

# = 2(-sinx)(cosx)#

# = -2sinxcosx#

# = -f(x)#

Since #f(-x) = -f(x)# for all #x# in the domain, #f# is odd.

Alternative

You could observe that #f(x) = 2sinxcosx = sin(2x)#, so

#f(-x) = sin(2(-x)) = sin(-2x) = -sin(2x) = -f(x)#