"Is this correct?" Yes, it is.
If f is any function, we can define the "even part" of f(x) to be and the "odd part" of f(x) to be [tex]f_o(x)= \frac{f(x)- f(-x)}{2}[/itex]. is an "even function", f(-x)= f(x), is an "odd functin", f(-x)= -f(x), and their sum is .
If f is an even function to start with, its "odd part" will be 0 and if f is an odd function, its "even part" will be 0. Rational functions that involve only even powers of x are "even functions" and rational functions that involve only odd powers of x are "odd functions.
Here, you were asked to find the even part of the function which is, as you finally found, . Its odd part is, of course, .
Here, you are basically asked