Answer:
Explanation:
To find the range of the function g(x)=∣x−1∣−1g(x)=∣x−1∣−1, we need to determine all possible values that g(x)g(x) can take.
First, let's analyze the function ∣x−1∣∣x−1∣. The absolute value of any real number is always non-negative. So, ∣x−1∣≥0∣x−1∣≥0 for all xx.
Next, we subtract 1 from ∣x−1∣∣x−1∣, which means the minimum value of g(x)g(x) occurs when ∣x−1∣∣x−1∣ is equal to 0.
To find the value of xx when ∣x−1∣=0∣x−1∣=0, we set the expression inside the absolute value bars to zero and solve for xx:
∣x−1∣=0∣x−1∣=0
x−1=0x−1=0
x=1x=1
Now, let's consider two cases for xx relative to 1:
When x<1x<1: In this case, x−1<0x−1<0, so ∣x−1∣=−(x−1)∣x−1∣=−(x−1). Thus, g(x)=−(x−1)−1=−x+2g(x)=−(x−1)−1=−x+2.
When x≥1x≥1: In this case, x−1≥0x−1≥0, so ∣x−1∣=x−1∣x−1∣=x−1. Thus, g(x)=(x−1)−1=x−2g(x)=(x−1)−1=x−2.
Now, we can see that for x<1x<1, g(x)=−x+2g(x)=−x+2, and for x≥1x≥1, g(x)=x−2g(x)=x−2.
So, the range of g(x)g(x) is (−∞,2](−∞,2] for x<1x<1 and [−2,∞)[−2,∞) for x≥1x≥1.
Combining both ranges, the overall range of g(x)=∣x−1∣−1g(x)=∣x−1∣−1 is [−2,∞)[−2,∞).
The correct option is:
D. [−2,∞)[−2,∞)