i need help, i don't understand the concept of odd and even functions, i dont get it, when a question asks which is an odd or even how do i know if its odd or even, after i substituted f(x) any help... help with an example, would be nice
ok even function are symmetric about the y axis: the condition is that f(-x) = f(x), hopefully this result seems fairly obvious, if you have something symmtric about the y axis, if i go out 3 units right from the origin and i go 3 units left fro the origin i will get the same function value;
e.g. Show f(x) = x^4 - x^2 is an even function
f(-x) = (-x) ^4 - ( -x)^2 = x^4 -x^2= f(x), hence even
for odd functions we have rotational symmtry about the origin, the condition to show is f(-x) = -f(x)
e.g. show f(x) = x^3 - x^5 is odd
f(-x) = (-x)^3 - (-x)^5 = -x^3 - (-x^5) = - [x^3 -x^5] = -f(x)
i know my spelling is shit
these formulas have a strong geometric meaning behind them, look at what f(-x)= -f(x) is actually saying.
its saying that if i go a distance "x" left from the origin and get the function value, it will equal the negative ( ie a reflection over the x axis) of the function value if i go the equal distance in the right direction. Just thinking about this geometrically it should be obvious that i can rotate the graph 180 degrees about the origin and get the same thing.
Maths is not just about chanting "f(-x) = - f(x) " , if you understand what the formula is saying, you will NEVER forget it.