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]
