PerenialQuest
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Hello all,
This is a question on a problem set for my Calculus 1 class. Please help if you can.
A function, f defined on the set of real numbers is said to have even symmetry if f(-x) = f(x) for all x, and is said to have odd symmetry if f(-x) = -f(x) for all x. Use the chain rule to verify that if a function has even symmetry, its derivative function has odd symmetry and vice versa. Note that I am asking you to show that this is true in general. You cannot simply cite examples, such as the fact that f(x) = x^2 has even symmetry and its derivative function f'(x) = 2x has odd symmetry. You must show that this relationship holds for all even and odd functions on the set of real numbers.
Chain rule: [d/dx] f(g(x)) = f'(g(x)) * g'(x)
I would love to attempt this, but I do not know where to begin. I see that this conjecture is clearly true in any specific example like the one mentioned above, but I do no know how to go about proving this in general. If someone can just suggest a strategy I can probably figure it out or I can at least try it and return with evidence of a concerted effort.
Thanks for the help. Richard
This is a question on a problem set for my Calculus 1 class. Please help if you can.
Homework Statement
A function, f defined on the set of real numbers is said to have even symmetry if f(-x) = f(x) for all x, and is said to have odd symmetry if f(-x) = -f(x) for all x. Use the chain rule to verify that if a function has even symmetry, its derivative function has odd symmetry and vice versa. Note that I am asking you to show that this is true in general. You cannot simply cite examples, such as the fact that f(x) = x^2 has even symmetry and its derivative function f'(x) = 2x has odd symmetry. You must show that this relationship holds for all even and odd functions on the set of real numbers.
Homework Equations
Chain rule: [d/dx] f(g(x)) = f'(g(x)) * g'(x)
The Attempt at a Solution
I would love to attempt this, but I do not know where to begin. I see that this conjecture is clearly true in any specific example like the one mentioned above, but I do no know how to go about proving this in general. If someone can just suggest a strategy I can probably figure it out or I can at least try it and return with evidence of a concerted effort.
Thanks for the help. Richard