Even and Odd Functions

Even Functions

If f(x)= f(-x) then the function is even.
This is basically saying that every time you input a number for x, that number will change its sign into the opposite. For example, you input a 1 it will change into -1. If you input -2, it will change to 2. Graphing wise, the function is symmetrical in respect of the y-axis. Every input will have a output that is a reflection of the original graph. The input is changed into negative hence it is (-x,y) instead of (x,y)

Look at the equation: f(x)= x^2

When x = 1, then f(1) = 1. Also when placed into f(-x) since f(x) = f(-x), f(-(1)) = 1. The point (-1,1) is the reflection of the point (1,1) from the original equation f(x)=x^2. Therefore the equation f(x)=x^2 is a even function.

Odd Functions

If f(- x)= - f(x) then the function is odd.
This is saying that the original function f(x) equals the negative function of itself. For example, if f(x) = x+3 then - f(x) = - (x+3). Graphing wise, the function is symmetrical according to the origin. Not only the output is changed to negative but the input as well. Instead of (x,y) it becomes (-x,-y)

Look at the equation: f(x) = x^3


When x = 2, then f(- 2) = - 8 and - f(2) = - 8. The point (-2,-8) is the reflection of (-2,8) in respect of the x-axis. Look back at f(x) = x^2. Its negative domain (x) is the reflection of the positive domain. In f(x) = x^3, its negative domain is the reflection of the negative domain of f(x) = x^2. The negative x portion of the graph is the reflection of the y-axis as well as the x-axis but the graph involves the point (o,o), the origin hence the function is odd.