To determine if a function is even or odd, you use the algebraic test by substituting −x-x−x for xxx in the function and comparing the result with the original function f(x)f(x)f(x):

• A function is even if f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) for all xxx in the domain. This means the function's graph is symmetric about the y-axis.
• A function is odd if f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) for all xxx in the domain. This means the function's graph is symmetric about the origin.
• If neither condition holds, the function is neither even nor odd.

In practice, replace xxx with −x-x−x in the expression, simplify, and check whether the result equals f(x)f(x)f(x) (even) or −f(x)-f(x)−f(x) (odd). If neither, it's neither even nor odd. This test works algebraically without needing the graph, but the graphical interpretation can also help: even functions mirror about the y-axis, odd functions have origin symmetry.