Even And Odd Functions

Symmetry simplifies the study of mathematical functions. Recognizing even and odd functions allows us to simplify graph sketching and evaluate definite integrals over symmetric intervals.

Quick Reference

Property	Algebraic Condition	Graphical Symmetry	Examples
Even Function	$f (- x) = f (x)$	Symmetric about y-axis	$x^{2}, \cos x, \| x \|$
Odd Function	$f (- x) = - f (x)$	Symmetric about the origin	$x^{3}, \sin x, 1 / x$

Even Functions

A function $y = f (x)$ on a symmetric interval $(- a, a)$ is even if:

f(-x) = f(x) for every x in (-a, a)

The graph of an even function is symmetric about the y-axis.

Graph showing y-axis reflection symmetry of an even function — An even function graph is symmetric with respect to the y-axis.

Odd Functions

A function $y = f (x)$ on a symmetric interval $(- a, a)$ is odd if:

f(-x) = -f(x) for every x in (-a, a)

The graph of an odd function is symmetric about the origin.

Graph showing 180 degree rotational symmetry about the origin for an odd function — An odd function graph is symmetric with respect to the origin.

Quick Reference

Even Functions

Odd Functions

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