Examples

  Clear[f]
  f[x_]:=x^3-3x^2-7x+5

  p1=Plot[f[x],{x,-4,4}];

  evenpart[f_,x_]:=1/2(f[x]+f[-x]);
  oddpart[f_,x_]:=1/2(f[x]-f[-x]);

  f[x]
  Simplify[evenpart[f,x]]
  Simplify[oddpart[f,x]]

  p2=Plot[evenpart[f,x],{x,-4,4}];

  p3=Plot[oddpart[f,x],{x,-4,4}];

  Show[p1,p2,p3];

If your function is already even, the odd part is 0. If the function is already odd, the even part is 0.

  g[x_]:=Sin[x]

  evenpart[g,x]
  oddpart[g,x]

  h[x_]:=Cos[x]

  evenpart[h,x]
  oddpart[h,x]

Up to Even and Odd Parts