- #1
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I'm puzzled by this question: Show that for all function f:R-->R. there exists an even function p and an odd function i such that f(x) = p(x) + i(x) forall x in R.
I got nothing.
I got nothing.
Proving the Existence of Even and Odd Functions in f:R-->R
In summary, for any function f:R-->R, there exists an even function p and an odd function i such that f(x) = p(x) + i(x) for all x in R. This can be proven by examining f(-x) and supposing that the result of the theorem is true, which would imply the existence of p and i such that f(-x) = p(x) - i(x). However, this is as far as the proof goes and further analysis is needed to fully prove the theorem.
- #2
Physics Monkey
Science Advisor
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Try looking at f(-x) and relating it to p(x) and i(x). Do you notice anything?
- #3
- 4,807
- 32
But there is nothing to look at. f(-x) = ......?
The only thing would be SUPPOSING the result of the thorem is true, then it would implies that there exist p and i such that f(x) = p+i and hence f(-x) = p(x)-i(x), but that's as far as that goes.
The only thing would be SUPPOSING the result of the thorem is true, then it would implies that there exist p and i such that f(x) = p+i and hence f(-x) = p(x)-i(x), but that's as far as that goes.
- #4
Hurkyl
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No it's not.
- #5
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Oh I see. That was very insightful Hurkyl. 
FAQ: Proving the Existence of Even and Odd Functions in f:R-->R
What is an even function?
An even function is a mathematical function where f(-x) = f(x) for all values of x. This means that the function is symmetric about the y-axis, and any point on the graph has a mirror image on the other side.
What is an odd function?
An odd function is a mathematical function where f(-x) = -f(x) for all values of x. This means that the function is symmetric about the origin, and any point on the graph has a reflection across the origin.
How do you prove that a function is even?
To prove that a function is even, you need to show that f(-x) = f(x) for all values of x. This can be done by substituting -x for x in the function and simplifying the equation. If the resulting equation is the same as the original function, then it is an even function.
How do you prove that a function is odd?
To prove that a function is odd, you need to show that f(-x) = -f(x) for all values of x. This can be done by substituting -x for x in the function and simplifying the equation. If the resulting equation is the negative of the original function, then it is an odd function.
Why is it important to prove the existence of even and odd functions?
Proving the existence of even and odd functions is important because it helps us understand the behavior of mathematical functions. It also allows us to apply certain properties and rules that are specific to even or odd functions, making it easier to solve problems and evaluate functions. Additionally, many real-world phenomena can be modeled using even and odd functions, so being able to identify them is crucial in scientific and mathematical analysis.