To determine which of the given statements accurately describes the function [tex]\( h(x) = -2 \sqrt{x-3} \)[/tex], let's analyze the function in detail.
1. Domain of the Function:
The expression inside the square root, [tex]\( x-3 \)[/tex], must be non-negative for [tex]\( \sqrt{x-3} \)[/tex] to be defined. Therefore, we have:
[tex]\[
x - 3 \geq 0 \implies x \geq 3
\][/tex]
This means the domain of [tex]\( h(x) \)[/tex] is [tex]\( x \geq 3 \)[/tex].
2. Behavior of the Square Root Function:
The square root function [tex]\( \sqrt{x-3} \)[/tex] is defined and increasing for all [tex]\( x \geq 3 \)[/tex].
3. Considering the Function [tex]\( h(x) \)[/tex]:
The function [tex]\( h(x) \)[/tex] is given by:
[tex]\[
h(x) = -2 \sqrt{x-3}
\][/tex]
Since the square root function [tex]\( \sqrt{x-3} \)[/tex] is increasing on its domain, multiplying it by [tex]\(-2\)[/tex] will reflect it over the x-axis and multiply by 2, making [tex]\( h(x) \)[/tex] a decreasing function.
4. Determine the Correct Interval:
From our analysis, [tex]\( h(x) \)[/tex] is defined for [tex]\( x \geq 3 \)[/tex] and decreases as [tex]\( x \)[/tex] increases.
Hence, the function [tex]\( h(x) \)[/tex] is decreasing on the interval [tex]\( (3, \infty) \)[/tex].
With this information, we can conclude that the correct statement describing [tex]\( h(x) \)[/tex] is:
[tex]\[
\boxed{\text{The function } h(x) \text{ is decreasing on the interval } (3, \infty) \, .}
\][/tex]
