Algebra/Chapter 11/Extrema and Continuity

Even and odd functions

Even functions

An even function is defined as a function $f$ such that $f(-x)=f(x)$.
Geometrically an even function can be defined as a function that exhibits a mirror image symmetry across the y-axis (the vertical line that passes through the origin).

An example of an even function is $h(x)=x^{2}$ because $f(5)=25=f(-5)$ and because $f(x)=x^{2}=f(-x)$ for all real numbers x.

Odd functions

An odd function is defined as a function $f$ such that $f(-x)=-f(x)$.
Geometrically an odd function can be defined as a function that exhibits a 180 degree rotational symmetry about the origin.

An example of an odd function is $f(x)=x^{3}$ because for all real numbers x, $f(x)=x^{3}=-((-x)^{3})=-f(-x)$ for example $f(2)=2^{3}=8=-((-2)^{3})=-(-8)=-((-2)^{3})=-f(-2)$