Determine all functions $f$ from the reals to the reals for which

(1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$, 

where $g(x)$ is the composition inverse function to $f(x)$.  (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)
To determine all functions \( f \) from the reals to the reals that satisfy the given conditions, we need to analyze the properties of \( f \) and its inverse \( g \).

1. **Given Conditions:**
   - \( f(x) \) is strictly increasing.
   - \( f(x) + g(x) = 2x \) for all real \( x \).
   - \( g(x) \) is the inverse function of \( f(x) \), i.e., \( f(g(x)) = x \) and \( g(f(x)) = x \).

2. **Express \( g(x) \) in terms of \( f(x) \):**
   Since \( g(x) \) is the inverse of \( f(x) \), we have:
   \[
   f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
   \]
   Using the given condition \( f(x) + g(x) = 2x \), we can solve for \( g(x) \):
   \[
   g(x) = 2x - f(x)
   \]

3. **Substitute \( g(x) \) back into the inverse condition:**
   Substitute \( g(x) = 2x - f(x) \) into \( f(g(x)) = x \):
   \[
   f(2x - f(x)) = x
   \]

4. **Analyze the functional equation:**
   To find \( f(x) \), assume a linear form \( f(x) = ax + b \). We need to determine \( a \) and \( b \) such that the conditions are satisfied.

5. **Substitute the linear form into the functional equation:**
   Let \( f(x) = ax + b \). Then:
   \[
   g(x) = 2x - (ax + b) = (2 - a)x - b
   \]
   Substitute \( g(x) \) into \( f(g(x)) = x \):
   \[
   f((2 - a)x - b) = a((2 - a)x - b) + b = x
   \]
   Simplify the equation:
   \[
   a(2 - a)x - ab + b = x
   \]
   Equate the coefficients of \( x \) and the constant terms:
   \[
   a(2 - a) = 1 \quad \text{and} \quad -ab + b = 0
   \]

6. **Solve the system of equations:**
   From \( a(2 - a) = 1 \):
   \[
   2a - a^2 = 1 \implies a^2 - 2a + 1 = 0 \implies (a - 1)^2 = 0 \implies a = 1
   \]
   From \( -ab + b = 0 \):
   \[
   b(-a + 1) = 0 \implies b(1 - 1) = 0 \implies b = 0
   \]

7. **Conclusion:**
   The only solution that satisfies all conditions is:
   \[
   f(x) = x
   \]

The final answer is \( \boxed{ f(x) = x } \).