What is how to tell if a function is even or odd?

A function can be classified as even, odd, or neither based on its symmetry. Here's how to determine the classification:

• Even Function: A function f(x) is even if f(-x) = f(x) for all x in its domain. Graphically, this means the function is symmetrical about the y-axis. See: Even Function

• Odd Function: A function f(x) is odd if f(-x) = -f(x) for all x in its domain. Graphically, this means the function has rotational symmetry of 180 degrees about the origin. See: Odd Function

• Neither: If a function doesn't satisfy either of the above conditions, it is neither even nor odd.

How to Check Algebraically:

1. Find f(-x): Replace every x in the function's equation with -x.

2. Simplify f(-x).

3. Compare f(-x) to f(x):

   • If f(-x) = f(x), the function is even.
   • If f(-x) = -f(x), the function is odd.
   • If f(-x) is not equal to either f(x) or -f(x), the function is neither even nor odd.

Examples:

• f(x) = x² is even because f(-x) = (-x)² = x² = f(x).
• f(x) = x³ is odd because f(-x) = (-x)³ = -x³ = -f(x).
• f(x) = x² + x is neither because f(-x) = (-x)² + (-x) = x² - x, which is not equal to f(x) or -f(x).