How do you determine whether #f(x)=(2x^5)-(2x^3)# is an odd or even function?

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Answer 1 · 2018-05-22T20:33:59

see below

the definition of an odd function is the characteristic #f(-x) = -f(x)#.

if you swap a certain #x#-value for its additive inverse, then the #y#-value will also change to its additive inverse.

here, #f(x) = 2x^5 - 2x^3#.

#f(-x)# would be the new expression when all the #x#s were swapped for #-x#s, so:

#f(-x) = 2(-x)^5 - 2(-x)^3#

#(-x)^5 = -x#
#(-x)^3 = -x#

hence, #f(-x) = -2x^5 - (-2x^3)#, or #-2x^5 + 2x^3#.

this is the negative of #f(x)#, which is #+2x^5 - 2x^3#.

for the function #f(x) = 2x^5 - 2x^3#, #f(-x) = -f(x)#.

therefore, #f(x) = 2x^5 - 2x^3# is an odd function.