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**Pn0yS0ld13r** Even and odd function are functions which satisfy some symmetry relation.

Geometrically, the graph of an even function is symmetric with respect to the y-axis. Therefore, let $\displaystyle f(x)$ be a real-valued function with a real variable. Then *f* is even if the following equation holds for all *x* in the domain of *f*.

$\displaystyle f(x) = f(-x)$.

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin; i.e., the graph remains unchanged after rotation of 180 degrees about the origin. Therefore, Therefore, let $\displaystyle f(x)$ be a real-valued function with a real variable. Then *f* is even if the following equation holds for all *x* in the domain of *f*.

$\displaystyle -f(x) = f(-x)$.

A function is neither even nor odd if it fails to hold for the above equations.

$\displaystyle f(x) = 2x+5$,

$\displaystyle f(-x) = -2x+5$, therefore $\displaystyle f(x) \neq f(-x)$ and $\displaystyle -f(x) \neq f(-x)$ so $\displaystyle f(x)$ is neither even nor odd.

$\displaystyle h(x) = -5x^{5}+3x^4$,

$\displaystyle h(-x) = 5x^{5}+3x^4$, therefore $\displaystyle h(x) \neq h(-x)$ and $\displaystyle -h(x) \neq h(-x)$ so $\displaystyle h(x)$ is also neither even nor odd.

**I finally understand. Thanks!**