Question

Determine whether the given function is even, odd, or neither.

f(x) = -3x⁵ + x³
a. Neither
b. Even
c. Odd
d. Cannot be determined

Answer 1

The function f(x) = -3x^5 + x^3 is neither even nor odd because f(-x) does not equal f(x) or -f(x), which means it does not display the necessary symmetry about the y-axis or the origin to be classified as even or odd.

Step-by-step explanation:

To determine whether the given function f(x) = -3x⁵ + x³ is even, odd, or neither, we need to evaluate the function at -x and compare it to f(x). For even functions, f(x) is equal to f(-x), which shows symmetry about the y-axis. For odd functions, f(x) is equal to -f(-x), displaying symmetry about the origin. Let's apply these definitions:

• 
  
• Compute f(-x): f(-x) = -3(-x)⁵ + (-x)³ = -3x⁵ - x³.
• 
  
• Compare with f(x): f(-x) is not equal to f(x) and it's also not equal to -f(x).

Since f(-x) is neither f(x) nor -f(x), the function is neither even nor odd. Therefore, the correct answer is neither.