Let's analyze the function [tex]\( h(x) = \log_{0.5} \left( \frac{z}{16} \right) \)[/tex] and understand its behavior to determine which statements are supported by the graph of this function.
### 1. Examining the Base of the Logarithm:
The function uses a logarithm with base [tex]\( 0.5 \)[/tex]. Recall that the logarithm base [tex]\( 0.5 \)[/tex] is decreasing because [tex]\( 0.5 < 1 \)[/tex]. This informs us that as the value inside the logarithm increases, the value of the logarithm decreases.
### 2. Behavior with respect to [tex]\( x \)[/tex]:
The function argument inside the logarithm [tex]\( \frac{z}{16} \)[/tex] depends on [tex]\( z \)[/tex], not directly on [tex]\( x \)[/tex]. However, let's consider different scenarios:
- Statement 1: "As [tex]\( x \)[/tex] increases, [tex]\( h(x) \)[/tex] decreases."
Since [tex]\( h(x) \)[/tex] is implicitly reflecting how the antibiotic concentration relates to time and given that this is a decreasing logarithmic function, this statement is valid. As the concentration [tex]\( x \)[/tex] increases, the required time [tex]\( h(x) \)[/tex] decreases because the effect is quicker in higher concentrations.
### 3. Checking Specific Values:
- Statement 2: "When [tex]\( z = 16 \)[/tex], [tex]\( h(x) = 0 \)[/tex]."
Substitute [tex]\( z = 16 \)[/tex] into the function:
[tex]\[
h(x) = \log_{0.5} \left( \frac{16}{16} \right) = \log_{0.5} (1)
\][/tex]
Since [tex]\(\log_{0.5}(1) = 0\)[/tex], the statement is valid. When [tex]\( z \)[/tex] equals 16, the logarithm evaluates to 0, supporting [tex]\( h(x) = 0 \)[/tex].
### 4. Domain and Function Definition:
- Statement 3: "The function is undefined for [tex]\( x \leq 0 \)[/tex]."
The logarithm function is only defined for positive arguments. [tex]\( \frac{z}{16} \)[/tex] must be greater than zero. If [tex]\( x \leq 0 \)[/tex], this would make the function undefined due to either negative arguments or division by zero/negative, which isn't valid for real logarithms.
Given these analyses, the valid statements supported by the graph are:
1. As [tex]\( x \)[/tex] increases, [tex]\( h(x) \)[/tex] decreases.
2. When [tex]\( z = 16, h(x) = 0.
3. The function is undefined for \( x \leq 0 \)[/tex].
Hence, the graph supports these three statements.
