Given a nonconstant polynomial with real coefficients $f(x),$ let $S(f)$ denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that $S(p) = 7,$ $S(q) = 9,$ and $S(p-q)= 11$. Find, with proof, all possible values for $S(p + q)$.
To solve the problem, we need to find the possible values of \( S(p + q) \) given that \( S(p) = 7 \), \( S(q) = 9 \), and \( S(p - q) = 11 \). We will use Vieta's formulas and properties of polynomial roots to derive the solution.

1. **Expressing the sums of roots using Vieta's formulas:**
   - For a polynomial \( p(x) \) of degree \( n \) with roots \( r_1, r_2, \ldots, r_n \), Vieta's formulas give us:
     \[
     S(p) = r_1 + r_2 + \cdots + r_n = -\frac{a_{n-1}}{a_n}
     \]
   - Similarly, for a polynomial \( q(x) \) of degree \( m \) with roots \( s_1, s_2, \ldots, s_m \), we have:
     \[
     S(q) = s_1 + s_2 + \cdots + s_m = -\frac{b_{m-1}}{b_m}
     \]
   - For the polynomial \( p(x) - q(x) \), the sum of the roots is:
     \[
     S(p - q) = t_1 + t_2 + \cdots + t_k = -\frac{c_{k-1}}{c_k}
     \]

2. **Given conditions:**
   - \( S(p) = 7 \)
   - \( S(q) = 9 \)
   - \( S(p - q) = 11 \)

3. **Analyzing the sum of roots of \( p(x) + q(x) \):**
   - Let \( p(x) \) and \( q(x) \) be polynomials of degrees \( n \) and \( m \) respectively.
   - The polynomial \( p(x) + q(x) \) will have roots that are the sums of the roots of \( p(x) \) and \( q(x) \).

4. **Using the given sums of roots:**
   - We know that:
     \[
     S(p) = 7, \quad S(q) = 9, \quad S(p - q) = 11
     \]
   - We need to find \( S(p + q) \).

5. **Using the properties of polynomial roots:**
   - Consider the polynomial \( p(x) \) and \( q(x) \) with the given sums of roots.
   - The sum of the roots of \( p(x) + q(x) \) can be expressed as:
     \[
     S(p + q) = S(p) + S(q) - S(p - q)
     \]

6. **Calculating \( S(p + q) \):**
   - Substitute the given values:
     \[
     S(p + q) = 7 + 9 - 11 = 5
     \]

Therefore, the only possible value for \( S(p + q) \) is \( \boxed{5} \).