Even Functions — Cheat sheet

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Key method

f(-x)=f(x).

Test the whole formula, not merely its visible exponents. Absolute value and cosine are even; many expressions containing odd powers are neither even nor odd.

Example

For $f(x)=x^4-3x^2+2$ , substitution gives $f(-x)=(-x)^4-3(-x)^2+2=f(x)$ . For $g(x)=x^2+x$ , $g(-x)=x^2-x$ is generally different. Restricting $x^2$ to $[0,\infty)$ removes the symmetric domain, so the restricted function is not even under the standard definition.

Avoid this mistake

Matching $f(1)$ and $f(-1)$ alone does not prove evenness; the equality must hold throughout the domain.