Even and Odd Functions

Even Functions

Hello everyone!

Mathematically, even functions are defined as f(x)=f(-x).

All even functions are symmetrical about the y-axis. This means that all points graphed in Quadrant 1 will be reflected in Quadrant 2 (& vice versa) and all points graphed in Quadrant 3 will be reflected in Quadrant 4 (& vice versa).

Some examples of even functions include f(x)= absolute value of x, f(x)=x^2, and f(x)=cosx.

For those visual people out there, the graphs of even functions look like so:

Example:

Consider the function f(x) = x^2.

Recall: For this function to be even, f(x)must equal f(-x). For this function to be odd, f(-x) must equal-f(x).

f(x)=x^2

f(-x)=(-x)^2= x^2

-f(x)= - (x^2)

In this case, f(x)=f(-x), therefore, function f is even.

Odd Functions

Mathematically, odd functions are defined as f(-x)=-f(x).

All odd functions are symmetrical about the origin. This means that all points graphed in Quadrant 1 will reflect like a mirror image in Quadrant 3. Also, all points graphed in Quadrant 2 will reflect like a mirror image in Quadrant 4.

Some examples of odd functions include: f(x)=1/x, f(x)=x^3, and f(x)=sinx.

Some visual representations of odd functions include:

Example:

Consider the function f(x) =x^3.

Recall: For this function to be even, f(x)must equal f(-x). For this function to be odd, f(-x) must equal-f(x).

f(x)=x^3

f(-x)=(-x)^3= -x^3

-f(x)= - (x^3)=-x^3

In this case, f(-x)=-f(x), therefore, function f is odd.

I hope this explains even and odd functions rather accurately. For further explanation, use the following link to watch a useful video i found on youtube. :]

http://www.youtube.com/watch?v=v2ni9bwO4v4