The equation you provided is:

\[
g(x) + g(2) = \frac{2x + 3}{x - 1}
\]

This is a functional equation, and to find \( g(x) \), we need to analyze it more closely. We know that \( g(x) \) is a function, and the equation relates \( g(x) \) to \( g(2) \).

### Step 1: Find \( g(2) \)
Substitute \( x = 2 \) into the equation:
\[
g(2) + g(2) = \frac{2(2) + 3}{2 - 1}
\]
\[
2g(2) = \frac{4 + 3}{1} = 7
\]
\[
g(2) = \frac{7}{2}
\]

### Step 2: Rewrite the original equation using \( g(2) = \frac{7}{2} \)
Now substitute this value of \( g(2) \) back into the original equation:
\[
g(x) + \frac{7}{2} = \frac{2x + 3}{x - 1}
\]

### Step 3: Solve for \( g(x) \)
Now isolate \( g(x) \):
\[
g(x) = \frac{2x + 3}{x - 1} - \frac{7}{2}
\]

To simplify, we need to combine the terms on the right-hand side. First, find a common denominator:
\[
g(x) = \frac{(2x + 3) \cdot 2}{2(x - 1)} - \frac{7(x - 1)}{2(x - 1)}
\]

This gives:
\[
g(x) = \frac{2(2x + 3) - 7(x - 1)}{2(x - 1)}
\]
\[
g(x) = \frac{4x + 6 - 7x + 7}{2(x - 1)}
\]
\[
g(x) = \frac{-3x + 13}{2(x - 1)}
\]

Thus, the function \( g(x) \) is:
\[
g(x) = \frac{-3x + 13}{2(x - 1)}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore further:
1. What happens to \( g(x) \) as \( x \to 1 \)?
2. How do you find the domain of \( g(x) \)?
3. What is the behavior of \( g(x) \) as \( x \to \infty \)?
4. Can you differentiate \( g(x) \) to find \( g'(x) \)?
5. How would the equation change if \( g(2) \) was a different value?

**Tip**: Always check for points where the function might be undefined by examining the denominator.

Explore how to solve the functional equation g(x) + g(2) = (2x + 3)/(x - 1) step-by-step. Learn key algebraic techniques such as substitution and rational expression simplification. Ideal for students in Grades 10-12.

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