The function you're asking to graph is $g(x) = \log_{0.5}(x)$. This is a logarithmic function with a base of 0.5. The parent function for logarithms is generally $\log_b(x)$, where $b$ is the base.

Key Characteristics of $g(x) = \log_{0.5}(x)$:

1. Domain: The function is only defined for $x > 0$.
2. Range: The range is all real numbers $(-\infty, \infty)$.
3. Intercept: It crosses the x-axis at $x = 1$ because $\log_{0.5}(1) = 0$.
4. Asymptote: There is a vertical asymptote at $x = 0$.
5. Behavior: Since the base $0.5$ is between 0 and 1, the function is decreasing. This means as $x$ increases, $g(x)$ decreases.

I will now graph this function.It seems there was an issue generating the graph. I will try again.It seems the graph is still not generating correctly. I will describe the graph instead:

Description of the Graph:

• Shape: The graph of $g(x) = \log_{0.5}(x)$ starts high on the left (as $x$ approaches 0 from the positive side) and decreases as $x$ increases, curving downward as it approaches the x-axis.
• Asymptote: The vertical asymptote at $x = 0$ means the graph goes infinitely down as it gets closer to the y-axis from the right.
• Intercept: The graph crosses the x-axis at $x = 1$, since $\log_{0.5}(1) = 0$.
• Decreasing Nature: The graph decreases as $x$ increases because the base of the logarithm is between 0 and 1.

Would you like to explore this further, or perhaps try another question?

Related Questions:

1. What is the general shape of a logarithmic function when the base is less than 1?
2. How does changing the base of a logarithm affect the graph?
3. What happens to the graph of $g(x) = \log_{b}(x)$ if $b > 1$?
4. How would the graph change if the function were $g(x) = -\log_{0.5}(x)$?
5. Can you explain why there is a vertical asymptote at $x = 0$?
6. How does the logarithm base relate to exponential functions?
7. What are some real-world applications of logarithmic functions?
8. How would you solve an equation involving $\log_{0.5}(x)$?

Tip:

When graphing logarithmic functions, always consider the base. Bases between 0 and 1 will yield decreasing functions, while bases greater than 1 will yield increasing functions.