Question

How do you determine if `f(x) = x^3 - x^7` is an even or odd function?

Answer 1

odd function

Explanation:

To determine if a function is even/odd consider the following.

• If f(x) = f( -x) , then f(x) is even

Even functions are symmetrical about the y-axis.

• If f( -x) = - f(x) , then f(x) is odd

Odd functions have symmetry about the origin.

Test for even

`f(-x)=(-x)^3-(-x)^7=-x^3+x^7`

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

Test for odd

`-f(x)=-(x^3-x^7)=-x^3+x^7`

Since f( -x) = - f(x) , then f(x) is odd.

Here is the graph of f(x). Note symmetry about origin.
graph{x^3-x^7 [-10, 10, -5, 5]}