The problem in the image can be summarized as follows:

You are given a rational function $f(x)$ such that: $f(x) + f\left( \frac{x-1}{x} \right) = 1 + x$ for some real numbers $x \neq 0$ and $x \neq 1$.

You are tasked with determining the value of: $f(10) + \frac{1}{180}$

Step 1: Analyze the given equation

Let's define $y = \frac{x-1}{x}$. Simplifying this expression, we get: $y = 1 - \frac{1}{x}.$ This suggests a transformation that can be used to analyze the function further.

To solve for $f(x)$, we would need to manipulate this equation to express $f(x)$ explicitly.

I will work through this equation now to find $f(x)$, then substitute $x = 10$ to find $f(10)$, and finally calculate $f(10) + \frac{1}{180}$.

Let me proceed with solving this systematically.It seems I encountered an issue with the function definition. I'll fix that and reattempt solving the equation step by step. Let's try defining the function and solving the equation properly now.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like a detailed explanation of how to approach solving this problem manually?

Here are five related questions to deepen your understanding:

1. What steps can be used to solve functional equations?
2. How can transformations help simplify functional equations like this one?
3. What are rational functions, and how do they behave with specific values?
4. How do you determine restrictions (like $x \neq 0$ or $x \neq 1$) in functional equations?
5. How can you verify the solution to a functional equation?

Tip: For complex functional equations, it's often useful to substitute specific values for $x$ to uncover patterns.