To determine the range of the function [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex], we must first understand the properties of the sine function.
1. Understanding the Sine Function Range:
- The sine function, [tex]\( \sin(\theta) \)[/tex], has a range of [tex]\([-1, 1]\)[/tex] for any angle [tex]\(\theta\)[/tex]. This means that [tex]\(\sin(\theta)\)[/tex] will always produce values between -1 and 1 inclusive.
2. Applying the Transformation:
- The given function is [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex]. Here, the expression inside the sine function, [tex]\( x - \pi \)[/tex], shifts the sine function along the x-axis, but it does not affect the range of values the sine function can produce. So, [tex]\(\sin(x - \pi)\)[/tex] will also have a range of [tex]\([-1, 1]\)[/tex].
3. Scaling and Shifting:
- Next, we consider the effect of multiplying the sine function by 2: [tex]\( 2 \sin(x - \pi) \)[/tex].
- The range of [tex]\( 2 \sin(x - \pi) \)[/tex] will be twice that of [tex]\( \sin(x - \pi) \)[/tex]. Therefore, we scale the range [tex]\([-1, 1]\)[/tex]:
- Multiplying -1 by 2 gives -2.
- Multiplying 1 by 2 gives 2.
- Hence, the range of [tex]\( 2 \sin(x - \pi) \)[/tex] is [tex]\([-2, 2]\)[/tex].
4. Adding the Constant Term:
- Finally, the function adds 1 to the scaled sine function: [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex].
- This shifts the entire range of [tex]\( 2 \sin(x - \pi) \)[/tex] vertically by 1.
- Adding 1 to -2 gives -1.
- Adding 1 to 2 gives 3.
- Thus, the range of [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex] is [tex]\([-1, 3]\)[/tex].
Therefore, the correct range of the function [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex] is [tex]\(\boxed{-1 \text{ to } 3}\)[/tex], corresponding to answer choice D.
