Is the function #f(x) = x^3 + x sin^2 x# even, odd or neither?

1 Answer
Aug 16, 2015

Answer:

#f(x)# is odd

Explanation:

A function is even if if exhibits the property #f(-x) = f(x)#
A function is odd if it exhibits the property #f(-x) = -f(x)#

Let check for #f(x)#:
#f(-x)#
#= (-x)^3 + (-x) sin^2 (-x)#
#=-x^3-xsin^2 x#
#=-f(x)#

Thus, #f(x)# is odd. You can confirm this by graphing. Since #x^3# is odd and #xsin^2 x# is odd, therefore #f(x)=x^3 + xsin^2 x# is odd.

graph{x*(sin(x))^2 [-10.32, 10.295, -5.155, 5.155]}