IDNLearn.com is your go-to platform for finding accurate and reliable answers. Discover reliable and timely information on any topic from our network of knowledgeable professionals.

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].


Sagot :

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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.