If (f(x)=x^3) and (g(x)=2x), what is the nature of ((f+g))?

Correct answer: Odd function

The direct answer is A: \(f+g\) is an odd function. First add the functions: \((f+g)(x)=x^3+2x\). To test oddness, replace \(x\) by \(-x\): \((f+g)(-x)=(-x)^3+2(-x)=-x^3-2x=-(x^3+2x)=-(f+g)(x)\). Therefore it satisfies the definition of an odd function. Option A is correct. Option B is wrong because an even function would satisfy \(h(-x)=h(x)\), which does not happen here. Option C is wrong because the exact oddness test succeeds. Option D is wrong because the expression changes with \(x\); it is not a fixed constant. Both \(x^3\) and \(2x\) are odd, and the sum of odd functions is odd. Memory cue: every term with an odd power and no constant term changes sign when \(x\) changes to \(-x\).