Given the function \( f(x) = -2 \log_{0.5}(2 - x) + 1 \), we can analyze it in terms of domain, range, and other properties.

### Step 1: Understanding the Logarithmic Function
The function contains a logarithmic term \(\log_{0.5}(2 - x)\). For a logarithm to be defined:
\[
2 - x > 0 \quad \Rightarrow \quad x < 2
\]
So, the domain of \( f(x) \) is \( (-\infty, 2) \).

### Step 2: Simplifying the Function
The function can be expressed as:
\[
f(x) = -2 \log_{0.5}(2 - x) + 1
\]

### Step 3: Analysis of the Logarithmic Part
To understand the behavior of \(\log_{0.5}(2 - x)\), recall that \(\log_{0.5}(y)\) is decreasing for positive \(y\) because the base \(0.5\) is less than 1.

Thus, as \(x\) increases towards 2 (from the left), \(2 - x\) decreases, and \(\log_{0.5}(2 - x)\) becomes larger in the negative sense (since it’s an increasing negative value). This means the function \( f(x) \) will decrease overall because of the negative sign and the multiplication by -2.

### Step 4: Range and Asymptotic Behavior
- As \( x \) approaches \( 2 \) from the left, \( f(x) \) decreases without bound since the logarithmic term grows without bound negatively (but becomes positive due to the negative multiplication).
- As \( x \) approaches \(-\infty\), \( f(x) \) tends towards \( 1 \) because \(\log_{0.5}(2 - x)\) tends towards \( \log_{0.5}(\infty) \), which is \( 0 \).

Therefore, the range of \( f(x) \) is \( (-\infty, 1) \).

### Final Form of the Function
The function \( f(x) \) is:
\[
f(x) = -2 \log_{0.5}(2 - x) + 1
\]

### Properties:
- **Domain:** \( (-\infty, 2) \)
- **Range:** \( (-\infty, 1) \)
- **Behavior:** Decreasing on \( (-\infty, 2) \)
- **Asymptotes:** None, but \( f(x) \) grows negatively without bound as \( x \) approaches \( 2 \) from the left.

Would you like more details or have any questions?

### Further Exploration Questions:
1. What is the inverse function of \( f(x) \)?
2. How does the function behave as \( x \) approaches \( 2 \) exactly?
3. What is the derivative of \( f(x) \)?
4. Can you find the critical points of \( f(x) \)?
5. What happens if the base of the logarithm is changed to a different value, such as 10?
6. How would the graph of \( f(x) \) look like?
7. What is the integral of \( f(x) \)?
8. How does the function compare to \( f(x) = -2 \log(2 - x) + 1 \)?

#### Tip:
Always consider the domain restrictions when dealing with logarithmic functions to avoid undefined expressions.

Explore the function f(x) = -2 log base 0.5 of (2 − x) + 1, including its domain, range, and behavior. Learn about logarithmic functions and their applications in advanced mathematics.

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