# Function Rules

$fg\left( x \right) =f\left( g\left( x \right) \right) \\ \\ ff\left( x \right) =f\left( f\left( x \right) \right) ={ f }^{ 2 }\left( x \right) \\ \\ gf\left( x \right) =g\left( f\left( x \right) \right) \\ \\ gg\left( x \right) =g\left( g\left( x \right) \right) ={ g }^{ 2 }\left( x \right)$

# Even And Odd Functions

An even function exists when $f\left( x \right) =f\left( -x \right)$ for all values of x.

The graph of an even function must be symmetrical about the y-axis.

Examples of even functions have been listed below:

$f\left( x \right) ={ x }^{ 2 }$

$f\left( x \right) ={ x }^{ 4 }$

$f\left( x \right) =cosx$

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An odd function exists when $f\left( x \right) =-f\left( -x \right)$ for all values of x.

Graphs of odd functions should have 180 degrees rotational symmetry about the origin (0,0).

Examples of odd functions are listed below:

$f\left( x \right) =x$

$f\left( x \right) ={ x }^{ 3 }$

$f\left( x \right) =sinx$

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Most functions are neither even or odd.

An example of a function that is neither even or odd would be:

$f\left( x \right) ={ x }^{ 2 }+\frac { 1 }{ x }$