Question

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

Answer 1

f(x) is an odd function.

Explanation:

To determine if f(x) is even/odd, consider the following.

• If f(x) = f( -x) , then f(x) is even `AAx" in the domain"`

Even functions have symmetry about the y-axis.

• If f( -x) = - f(x) , then f(x) is odd `AAx" in the domain"`

Odd functions have half-turn symmetry about the origin.

Test for even

`f(-x)=(-x)^3-(-x)^5=-x^3+x^5≠f(x)`

Since f(x) ≠ f( -x) , then f(x) is not even.

Test for odd

`-f(x)=-(x^3-x^5)=-x^3+x^5=f(-x)`

Since f( -x) = - f(x) , then f(x) is odd.
graph{x^3-x^5 [-10, 10, -5, 5]}