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# Function symmetry introduction

Functions can be symmetrical about the y-axis, which means that if we reflect their graph about the y-axis we will get the same graph. There are other functions that we can reflect about both the x- and y-axis and get the same graph. These are two types of symmetry we call even and odd functions. Created by Sal Khan.

## Want to join the conversation?

• i noticed that even and odd also corresponded with the exponent in all the graphs equations... coincidence? i think not!!! but seriously can anyone answer my question?
• In the beginning, the people saw that functions like x^2 x^4 x^6 x^8 behaved like "even functions" so they called them so. But there are other functions that behave like even functions ( cos(x) ) but don't have an even exponent.
So this is not hard and fast rule that all even functions are going to have even numbers as exponents and all functions with even numbers with exponents are going to be even functions.
• why is math so complicated?
• Math is complicated to many of us because it uses many abbreviated symbols. And math in the U.S., at least, the higher math has been taught rapidly covering far too many topics for most of us to master in such a short time. We get very little time to focus on learning, so much of our time is spent in trying to memorize what we can. Our frustration mounts when we don't have time to practice what we learn. We miss out on understanding higher concepts when we have not mastered the lower concepts. Because we are behind we often do not know how to solve problems and do not get to see the beauty of applying them to our lives. Out of desperation to make sense we give up and see math as more complicated than it ought to be. Learning takes time, practice takes more time, and mastery of math takes much longer. In other countries where math is taught slower over fewer topics the students can see how to solve problems and they get abundant practice on the topics that are presented,
• How can you determine whether a function is even, odd, or neither by using just the equation of the function and not by graphing it?
• If for all x, f(x) = f(-x) then f(x) is even.
If for all x, f(x) = -f(-x) then f(x) is odd.
• At Sal said parabola, what does that mean?
• A parabola is basically a line that curves in a U-shape and is symmetrical down the middle of the U (vertically). That's how I think of it anyway.
• I don't whether my question is worth asking or not but what is the real life use of even and odd functions?
• So if the graph is symmetry to the y-axis, it is an even function. If the graph is symmetry to the x-axis it is an odd function?
• Not quite. For something to be an odd function, it has to have symmetry to the origin, not the x-axis. This means that if it has a point like (a, b), it also has the point (-a, -b). For example, y = x is an odd function because it does this.
• Can there be a function that is both odd and even?
• An odd function satisfies f(-x)= -f(x). An even function satisfies f(-x)=f(x). We want a function that does both of these. So we get -f(x)=f(x).
0=2f(x)
0=f(x)

So the only function that is both odd and even is the constant function 0.
• Interesting concepts and great video!

I was just wondering: is a circle centered on the origin both an even and odd function?

Ex: x^2 + y^2 = 1

Thanks!