4. If ( f: R rightarrow R ) such that ( f(x+1)=frac{2 f(x)-1}{f(x)-2} forall x in R ), then which of the following must be true? a) ( f(2011)=f(2014)^{x} ) b) ( f(2011)=f(2013) ) c) ( f(2012)=f(2020) ) d) ( f(2012)=f(2015) times )

We are given a functional equation \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that

\[
f(x + 1) = \frac{2f(x) - 1}{f(x) - 2} \quad \text{for all } x \in \mathbb{R}.
\]

We need to determine which of the following must be true:

a) \( f(2011) = f(2014)^x \)

b) \( f(2011) = f(2013) \)

c) \( f(2012) = f(2020) \)

d) \( f(2012) = f(2015) \)

### Step 1: Analyze the Functional Equation

First, let's analyze the given functional equation to observe any periodicity or specific values of \( f(x) \).

Let \( f(x) = y \). Then:

\[
f(x + 1) = \frac{2y - 1}{y - 2}.
\]

To understand the behavior of the function, let's iterate this equation.

### Step 2: Compute \( f(x + 2) \)
Substitute \( f(x + 1) \) into the functional equation:

\[
f(x + 2) = \frac{2f(x + 1) - 1}{f(x + 1) - 2}.
\]
Substituting \( f(x + 1) = \frac{2f(x) - 1}{f(x) - 2} \):

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

Simplify this:

\[
f(x + 2) = \frac{\frac{4f(x) - 2 - (f(x) - 2)}{f(x) - 2}}{\frac{2f(x) - 1 - 2(f(x) - 2)}{f(x) - 2}} = f(x).
\]

This shows that \( f(x + 2) = f(x) \), indicating that the function \( f(x) \) is periodic with a period of 2.

### Step 3: Evaluate the Options

Given that \( f(x + 2) = f(x) \), we can deduce the following:

- \( f(2011) = f(2013) \) because \( f(x) \) repeats every 2 units. (Option b)
- \( f(2012) = f(2014) \), so \( f(2012) = f(2020) \). (Option c)
- \( f(2012) = f(2014) = f(2016) = f(2018) = f(2020) \).

### Conclusion

- **Option b** \( f(2011) = f(2013) \) is true.
- **Option c** \( f(2012) = f(2020) \) is true.

The correct answers are **b) \( f(2011) = f(2013) \)** and **c) \( f(2012) = f(2020) \)**.

Would you like any further details or explanations?

### Related Questions
1. How does the periodicity of functions affect their properties?
2. Can \( f(x + n) = f(x) \) be solved for other forms of \( n \)?
3. What are the conditions under which a function becomes periodic?
4. How would the function behave if it were non-periodic?
5. What other forms of functional equations exhibit periodic solutions?
6. Can the period of a function be non-integer? If so, how?
7. How does the functional equation change if different constants are used?
8. What if \( f(x) \) were a trigonometric function, how would the solution differ?

**Tip:** When working with functional equations, it's important to test small values and look for patterns like periodicity that can simplify the analysis.

Explore the solution to a functional equation where f(x + 1) = (2f(x) - 1) / (f(x) - 2) for all real x. Understand the concept of periodicity in functions and its implications. Suitable for advanced mathematics enthusiasts.

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