EDGE Math 8

Even and Odd Functions

Even Functions
A function f is even if $f (x) = f (- x)$ for all x in the domain of f. Note that for an even function, the right side of the equation does not change if x is replaced by $- x .$

The given function f in the previous page, which is $f (x) = x^{2} - 1,$ is an example of an even function. To get the value of $f (- x),$ substitute $- x$ for x. You will get the following:

f (- x) = {(- x)}^{2} - 1 = x^{2} - 1.

Thus, $f (x) = f (- x) .$ This means that f is an even function.

Odd Functions
A function f is odd if $f (- x) = - f (x)$ for all x in the domain of f. Note that for an odd function, the sign of every term on the right side of the equation changes when x is replaced by $- x .$

The function $f (x) = x^{3} + x$ is an example of an odd function. To get the value of $f (- x),$ substitute $- x$ for x. You will get the following:

f (- x) = {(- x)}^{3} + (- x)

= - x^{3} - x

= - (x^{3} + x)

Thus, $f (- x) = - f (x) .$ This means that f is an odd function.