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Tool to check the parity of a function (even or odd functions): it defines its ability to verify symmetrical relations.

Answers to Questions

How to check if a function is even?

A function is even if the equality $$ f(x) = f(-x) $$ is true for all x from the domain of definition.

Example: Consider \( f(x) = x^2 \) in \( \mathbb{R} \), then \( f(-x) = (-x)^2 = x^2 = f(x) \), so \( f(x) \) is even.

Graphically, this involves that opposed abscissae have the same ordinates, this means that the ordinate axis is an axis of symmetry of the curve representing f.

How to check if a function is odd?

A function is odd if the equality $$ f(x) = -f(-x) $$ is true for all x from the domain of definition.

Example: Consider \( f(x) = x^3 \) in \( \mathbb{R} \), then \( -f(-x) = -(-x)^3 = x^3 = f(x) \), so \( f(x) \) is odd.

Graphically, this involves that opposed abscissae have opposed ordinates, this means that the origin (central point) (0,0) is a symmetry center of the curve representing f.

NB: if an odd function is defined in 0, then the curve passes at the origin: $$ f(0) = 0 $$

Why are functions called even or odd?

Developments in convergent power series or polynomials of even (respectively odd) functions have even degrees (respectively odd).

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