addisonbosley2
Answered

Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

What are the domain and range of [tex]g(x)=-3 \sqrt{x-1}[/tex]?

A. Domain: [tex](1, \infty)[/tex] and Range: [tex](0, \infty)[/tex]

B. Domain: [tex][-3, \infty)[/tex] and Range: [tex][0, \infty)[/tex]

C. Domain: [tex](-3, \infty)[/tex] and Range: [tex](-\infty, 0)[/tex]

D. Domain: [tex][1, \infty)[/tex] and Range: [tex](-\infty, 0][/tex]

Sagot :

GinnyAnswer
To determine the domain and range of the function [tex]\( g(x) = -3 \sqrt{x-1} \)[/tex], we will examine the conditions required for the function to be defined and then find the range based on the behavior of the function within its domain.

### Finding the Domain

The domain of a function consists of all the values of [tex]\( x \)[/tex] for which the function is defined. The function involves a square root, [tex]\( \sqrt{x-1} \)[/tex], which is only defined for non-negative arguments. Therefore, we need the expression inside the square root to be non-negative:

[tex]\[ x - 1 \geq 0 \][/tex]

Solving this inequality:

[tex]\[ x \geq 1 \][/tex]

So, the function [tex]\( g(x) \)[/tex] is defined for all [tex]\( x \geq 1 \)[/tex]. Hence, the domain of [tex]\( g(x) \)[/tex] is:

[tex]\[ [1, \infty) \][/tex]

### Finding the Range

Next, we determine the range of [tex]\( g(x) = -3 \sqrt{x-1} \)[/tex], which is the set of all possible output values (or [tex]\( g(x) \)[/tex] values).

1. Identify the minimum value of [tex]\( g(x) \)[/tex]:
- The minimum value of [tex]\( g(x) \)[/tex] occurs when the argument of the square root is minimized, i.e., when [tex]\( x = 1 \)[/tex].
- Substitute [tex]\( x = 1 \)[/tex] into the function:

[tex]\[ g(1) = -3 \sqrt{1 - 1} = -3 \sqrt{0} = 0 \][/tex]

So, the function reaches the value [tex]\( 0 \)[/tex] when [tex]\( x = 1 \)[/tex].

2. Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] increases, [tex]\( x - 1 \)[/tex] also increases, and thus [tex]\( \sqrt{x - 1} \)[/tex] increases.
- However, since it is multiplied by [tex]\(-3\)[/tex], the function value becomes more negative as [tex]\( x \)[/tex] increases:

[tex]\[ g(x) = -3 \sqrt{x - 1} \to -\infty \text{ as } x \to \infty \][/tex]

Hence, the output values can be arbitrarily large negative numbers, but the maximum output value (least negative) when [tex]\( x = 1 \)[/tex] is [tex]\( 0 \)[/tex].

Therefore, the range of [tex]\( g(x) \)[/tex] is:

[tex]\[ (-\infty, 0] \][/tex]

### Conclusion

Given the domain and range we have found:

- The domain of [tex]\( g(x) = -3 \sqrt{x-1} \)[/tex] is [tex]\( [1, \infty) \)[/tex]
- The range of [tex]\( g(x) = -3 \sqrt{x-1} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex]

The correct choice from the options given is:

D: [tex]\( [1, \infty) \)[/tex] and [tex]\( (-\infty, 0] \)[/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.