Given the summation of a function, is it possible to find the summation of a transformation of that function?

Question

For example, assume that I have a formula: $$S(n)=\sum_{x = 1}^n f(x)$$

Is it possible to find an expression for $$Q(n) = \sum_{x = 1}^n f(-x)$$ in terms of S(n) or would this need to be considered on an individual basis for each series.

Answer 1

Take the following function:

$$f(x)=\begin{cases}e^x \ \ x < 0 \\ 2x \ x \geq 0\end{cases}$$

As you can see, $f(-x)$ has nothing to do with $f(x)$. Functions like these are why this needs to be handled on a function-by-function basis.

Answer 2

It depends on the actual function, but if it arbitrary then there is no way to be able to relate them.