Even and Odd Functions

Even Functions: For even functions, first we need to know what f(-x)=f(x). In words, that means that for every input value of x, its output is the same as its opposite input. Simply stated, both x and -x have the same y value. Graphically, whatever shape the graph takes on one quadrant will be reflected across the y-axis and onto the other quadrant. Naturally, the graph will look symmetrical, divided along the y-axis. My explanation is a little confusing, so a good example to understand even functions would be be the function f(x)=x^2. No matter what x value is entered, its opposite x has the same y value. If you look at its graph, it is clearly symmetrical about the y-axis.

Odd Functions: Odd functions are harder to explain, so bare with me in my not-so-great-explanation. Again, we must first understand the mathematical meaning before we can understand its graphical meaning. The mathematical meaning, f(-x)=-f(x), means that if a point (x,y) lies on a graph, then the point (-x,-y) also lies on the graph. Or, if (x,-y) lies on a graph, the (-x,y) does too. Yes, I know, not the greatest explanation in the world. An easier way to think about it would be to say that if a point lies on a graph, then the opposite of that point (opposite x and y value) also lie on that point. Graphically, it's easier to understand. Graphically, a function is odd if the graph remains the same after rotating it around the origin 180 degrees (half a circle). Yes, that's really as simple as I can explain it. A perfect example would be the function f(x)=x. No matter what point you input, its opposite point is also a solution. It also looks exactly the same if rotated 180 degrees around the origin as well.

(Note: If any of my explanations were confusing, please let me know. I need to get better at explaining things to people.)