To determine if a function f(x)f(x)f(x) is even, odd, or neither, you use the following algebraic tests:

• Even Function: Substitute −x-x−x for xxx in the function. If the result equals the original function, i.e.,

f(−x)=f(x)f(-x)=f(x)f(−x)=f(x)

for all xxx in the domain, the function is even. Even functions have symmetry about the y-axis.

• Odd Function: Substitute −x-x−x for xxx and check if the result is the negative of the original function, i.e.,

f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x)

for all xxx in the domain. Odd functions have symmetry about the origin.

• Neither: If neither condition holds, the function is neither even nor odd.

How to apply the test:

1. Take the given function f(x)f(x)f(x).
2. Replace every xxx with −x-x−x to get f(−x)f(-x)f(−x).
3. Simplify f(−x)f(-x)f(−x).
4. Compare f(−x)f(-x)f(−x) with f(x)f(x)f(x) and −f(x)-f(x)−f(x):
   • If f(−x)=f(x)f(-x)=f(x)f(−x)=f(x), the function is even.
   • If f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x), the function is odd.
   • Otherwise, it is neither.

Example:

For f(x)=x5−3x3f(x)=x^5-3x^3f(x)=x5−3x3:

• Compute f(−x)=(−x)5−3(−x)3=−x5+3x3f(-x)=(-x)^5-3(-x)^3=-x^5+3x^3f(−x)=(−x)5−3(−x)3=−x5+3x3.
• Notice f(−x)=−(x5−3x3)=−f(x)f(-x)=-(x^5-3x^3)=-f(x)f(−x)=−(x5−3x3)=−f(x).
• So, f(x)f(x)f(x) is odd.

For f(x)=5x2−x4f(x)=5x^2-x^4f(x)=5x2−x4:

• Compute f(−x)=5(−x)2−(−x)4=5x2−x4f(-x)=5(-x)^2-(-x)^4=5x^2-x^4f(−x)=5(−x)2−(−x)4=5x2−x4.
• Notice f(−x)=f(x)f(-x)=f(x)f(−x)=f(x).
• So, f(x)f(x)f(x) is even.

This method works for any algebraic function and helps identify symmetry without graphing