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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 :

GinnyAnsweridn
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]
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