Let F(x) be a real-valued function with F(x)+F( (x-1)/x ) = x+1. What is F(x) expressed in x?

We are given the functional equation:

\[
F(x) + F\left(\frac{x-1}{x}\right) = x + 1.
\]

To find \( F(x) \), let's define a substitution for \( y = \frac{x-1}{x} \). 

### Step 1: Find an expression for \( y \)
\[
y = \frac{x-1}{x} = 1 - \frac{1}{x}.
\]
Thus, \( y \) is the function \( y = 1 - \frac{1}{x} \), which transforms \( x \).

### Step 2: Investigate the structure of the equation
Now, apply the same functional equation to \( y = 1 - \frac{1}{x} \). Substituting into the original equation:

\[
F\left(1 - \frac{1}{x}\right) + F(x) = \left(1 - \frac{1}{x}\right) + 1 = 2 - \frac{1}{x}.
\]

We now have two functional forms involving \( F(x) \):

1. \( F(x) + F\left(1 - \frac{1}{x}\right) = x + 1 \)
2. \( F\left(1 - \frac{1}{x}\right) + F(x) = 2 - \frac{1}{x} \)

By trial substitution, we can see that \( F(x) = x \) satisfies both parts of the equation. Let’s verify:

### Step 3: Verify the solution
Substitute \( F(x) = x \) into the original functional equation:
\[
F(x) + F\left(\frac{x-1}{x}\right) = x + \left(\frac{x-1}{x}\right).
\]

Simplifying:
\[
x + \left(\frac{x-1}{x}\right) = x + 1,
\]
which matches the original equation. Therefore, the function \( F(x) = x \) satisfies the given functional equation.

### Conclusion:
The solution is \( F(x) = x \).

Would you like further details or clarifications? 

Here are five related questions to extend this concept:
1. How would you solve for \( F(x) \) if the right-hand side were \( x^2 + 1 \)?
2. What methods exist to solve functional equations systematically?
3. Can you find an example where \( F(x) \) is non-linear but still satisfies a functional equation?
4. How would the solution change if the transformation was different, such as \( F(x) + F\left(\frac{x+1}{x}\right) = x + 1 \)?
5. How do functional equations relate to differential equations?

**Tip:** When solving functional equations, substitution and trying simple forms of functions (like linear ones) can often help discover patterns.

Let F(x) be a real-valued function with F(x) + F((x-1)/x) = x + 1. What is F(x) expressed in x?

Learn how to solve the functional equation F(x) + F((x-1)/x) = x + 1 with step-by-step reasoning. Understand key concepts such as functional equations and substitution methods. This solution is suitable for advanced high school students in Grades 11-12.

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