Writing a non symmetrical function as an even or odd function

Question

I have been given the task to rewrite a function as a sum of an even and an odd function. But when I went to analyze the parts of the original function I noticed that one part was neither even nor odd.

The original function was: $g(x) = (x + 1) / (x^2 - 3x + 4$)

After analyzing I realized that when using f(-x) to determine symmetry it produced a 'neither' result. How would I go about writing a function that is neither even nor odd as an even function. (It would have to be an even function as the (x+1) part of the original function is odd and since I need to write it as a sum of even and odd, the denominator would need to be even.

Answer 1

You don’t need to analyze components of a function $g(x)$ individually to write it as sum of an even and an odd function.

Any function $g(x)$ (of course assuming the domain is symmetric) can be written as: $$g(x)=\underbrace{\frac{g(x)+g(-x)}{2}}_{\text{even}}+\underbrace{\frac{g(x)-g(-x)}{2}}_{\text{odd}}$$ The first function is even and the second is odd.

Answer 2

Suppose we have a general function $f(x)$ and we want to express it as the sum of an even function and an odd function so that $$f(x)=E(x)+O(x)$$How might we find $E(x)$ and $O(x)$? Well one obvious thing would be to compute $$f(-x)=E(-x)+O(-x)=E(x)-O(x)$$ (using the definitions of even and odd)

Simply adding the two equations gives $$2E(x)=f(x)+f(-x): E(x)=\frac{f(x)+f(-x)}2$$

and subtracting gives $$2O(x)=f(x)-f(-x): O(x)=\frac{f(x)-f(-x)}2$$

So if $E(x)$ and $O(x)$ exist, they must have this form, and it is easy to check that they are even and odd functions respectively.

So the decomposition into even and odd functions, where it exists (depends where $f(x) is defined), is unique, and can be determined by these equations.