Given [tex]h(x) = -2 \sqrt{x - 3}[/tex], which of the following statements describes [tex]h(x)[/tex]?

A. The function [tex]h(x)[/tex] is increasing on the interval [tex](-\infty, 3)[/tex].
B. The function [tex]h(x)[/tex] is increasing on the interval [tex](-3, \infty)[/tex].
C. The function [tex]h(x)[/tex] is decreasing on the interval [tex](-\infty, 3)[/tex].
D. The function [tex]h(x)[/tex] is decreasing on the interval [tex](3, \infty)[/tex].



Answer :

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]