To determine the range of the function [tex]\(g(x) = 3|x - 1| - 1\)[/tex], we need to understand how the function behaves for all possible values of [tex]\(x\)[/tex].
Here is a step-by-step analysis of the function [tex]\(g(x)\)[/tex]:
1. Starting with the expression inside the absolute value:
[tex]\(|x - 1|\)[/tex]
The absolute value function [tex]\(|x - 1|\)[/tex] always yields a non-negative result (i.e., [tex]\(|x - 1| \geq 0\)[/tex]).
2. Scaling and translating the absolute value:
We multiply the absolute value by 3:
[tex]\[
3|x - 1|
\][/tex]
This multiplication scales the values of [tex]\(|x - 1|\)[/tex] by a factor of 3, but it remains non-negative (i.e., [tex]\(3|x - 1| \geq 0\)[/tex]).
3. Subtracting 1:
Subtracting 1 from [tex]\(3|x - 1|\)[/tex]:
[tex]\[
3|x - 1| - 1
\][/tex]
This shifts the whole graph of the function 1 unit downward.
4. Finding the minimum value:
There is a critical point when [tex]\(|x - 1| = 0\)[/tex]. This occurs when [tex]\(x = 1\)[/tex]. At [tex]\(x = 1\)[/tex]:
[tex]\[
g(1) = 3|1 - 1| - 1 = 3 \cdot 0 - 1 = -1
\][/tex]
So, the minimum value of [tex]\(g(x)\)[/tex] is [tex]\(-1\)[/tex].
5. Behavior as [tex]\(x\)[/tex] moves away from 1:
As [tex]\(x\)[/tex] moves away from 1 in either direction (to the left or right), the value of [tex]\(3|x - 1|\)[/tex] increases. Hence, [tex]\(3|x - 1| - 1\)[/tex] increases without bound. When [tex]\(x\)[/tex] goes to [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex], the value of [tex]\(3|x - 1| - 1\)[/tex] also tends to [tex]\(\infty\)[/tex].
6. Conclusion:
Given the minimum value of [tex]\(-1\)[/tex] and the function increasing without bound, the range of the function [tex]\(g(x)\)[/tex] is:
[tex]\[
[-1, \infty)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{B. \, [-1, \infty)}
\][/tex]
