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.