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

1 Answer
Aug 16, 2015

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]}