John O' Meara said:Homework Statement
Suppose that the function f has domain all real numbers. Show that f can be written as the sum of an even function and an odd function.
Homework Equations
f(-x) = f(x) is even and f(-x)=-f(x) is odd
The Attempt at a Solution
If g(x) is an even function it can be written as [tex]g(x) = \frac{f(x)+f(-x)}{2}[/tex] where f(x) is even and if h(x) is odd it can be written as h(x)=\frac{f(x)-f(-x)}{2} [/tex]. but how do you write f = g + h?
John O' Meara said:How did Hallsofivy get five terms in the last equation divided by 2. I would only have got four divided by 2.
John O' Meara said:How did Hallsofivy get five terms in the last equation divided by 2. I would only have got four divided by 2.
Yes, my eyes went squoggly for a moment. Thanks, n!kofeyn.n!kofeyn said:It looks like a typo. The second [itex]+f(-x)[/itex] shouldn't be there.
A sum of odd and even functions is a mathematical concept where two functions, one being odd and the other being even, are added together to form a new function. These functions have specific properties that make them behave differently when added together.
An odd function is symmetric about the origin and has the property that f(-x) = -f(x), while an even function is symmetric about the y-axis and has the property that f(-x) = f(x). Additionally, the sum of an odd and even function is always an odd function.
To determine if a function is odd or even, you can use the symmetry properties mentioned above. If f(-x) = -f(x), then the function is odd. If f(-x) = f(x), then the function is even. You can also look at the power of the variables in the function, as odd functions typically have an odd power and even functions have an even power.
One example of a sum of odd and even functions is f(x) = x^3 + x^2. The first term, x^3, is an odd function, and the second term, x^2, is an even function. When added together, they form a new function that is odd.
Understanding sum of odd and even functions is important in various areas of mathematics, such as calculus, differential equations, and Fourier series. These concepts are also used in physics and engineering to model and solve real-world problems. Additionally, they provide a deeper understanding of mathematical symmetry and how functions behave when combined.