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

Question

Sadie Ackerman · Accepted Answer

odd function

Examine the following to see if a function is even or odd. • f(x) is even if f(x) = f( -x) About the y-axis, even functions are symmetrical. • f(x) is odd if f(-x) = - f(x). There is symmetry around the origin of odd functions. Check for even #f(-x)=(-x)^3-(-x)^7=-x^3+x^7# f(x) is not even since f(x) ≠ f(-x). Check for odd #-f(x)=-(x^3-x^7)=-x^3+x^7# f(x) is odd since f(-x) = - f(x). See the symmetry surrounding the origin in this graph of f(x): graph{x^3-x^7 [-10, 10, -5, 5]}

Elijah Allen · Answer

undefined To determine if $ f(x) = x^3 - x^7 $ is an even or odd function:

1. Even Function: If $ f(-x) = f(x) $ for all $ x $ in the domain, then the function is even.
2. Odd Function: If $ f(-x) = -f(x) $ for all $ x $ in the domain, then the function is odd.

Apply these conditions to $ f(x) = x^3 - x^7 $:
- Substitute $ -x $ into $ f(x) $ to test for evenness.
- Substitute $ -x $ into $ -f(x) $ to test for oddness.
- Check if either condition holds true for all $ x $ in the function's domain.

After these tests, determine whether the function is even, odd, or neither.