Even Functions
the first definition results are the same as the second definition because the first definition (Graph symmetrical about the y axis) shows a change in the x corresponding to its y. y has two x as shown f(-x) = f(x) which has one corresponding y. Mathematically f(-x) and f(x) will yield the same output if an Evin function.
ex. f(x) = x2 + |x|
1) f(2) = 4+2; f(2) = 6
2) f(-2) = (-2)2 + |-2|; f(-2) = 4+2; f(-2) = 6
f(2) = f(-2)
Odd Functions
the mathematical definition results are the same as the first definition because when x is multiplied by "-". this means when x becomes its opposites value y also becomes its opposite value. in a graph this is shown as y going to the opposite direction when x is multiplied by "-". But zero has to be the origin. otherwise the distance from f(-x), -f(x) is the same. if zero is not the origin, the function can include the point of origin to make f(-x), -f(x) have the same value.
let o = origin
f(-x+x0) = (equation - y0)
-f(x+x0) = -(equation - y0)ex.
i think
ex. equation f(x)=x3+1
origin = (0,1)
f(-1-0)=(-1)3+1-1
f(-1)=-1
-f(1-0)=-((1)3+1-1)
-f(1)=-1
f(-1)=-f(1)
can graph my equations click here
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omg the last line of the even functions just made everything "click"!
ReplyDelete"f(2) = f(-2)"
f(-x)=f(x)
x = 2,-2
y=y, they will have the same y! :D
"this means when x becomes its opposite value y also becomes its opposite value."
ReplyDeleteah, the words i was trying to look for when i wrote my answer on odd functions. good explanation.
hmm comprehensive indeed.
ReplyDeletethis makes sense!:]
wow you my friend are good at explaining things... i think if you would have added some visuals it would have made it far easier to understand... for me at least, but good stuff sir
ReplyDeletehaha. I wholly agree w/ Jesus. I was going to quote the same thing! What is this whole origin stuff though? I dont think I understand.
ReplyDeletei see what you are talking about. when i did the origin stuff i was thinking plugging in numbers. if the equation f(x)= (x^3)+1 is an odd function, then -f(x) should be equal to f(-x), but if you do the math f(-1) = 0, while -f(1)= 2; not the same, but there the same distance apart: 1. which makes it symmetrical to the origin. i think. the graph moves up one and it is a cube. a cube is symmetrical. i might be wrong u tell me.
ReplyDelete