To determine the range of the function [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex], follow these steps:
1. Understand the Sine Function Range:
The sine function, [tex]\(\sin(x)\)[/tex], oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] for all real numbers [tex]\(x\)[/tex]. This means:
[tex]\[
-1 \leq \sin(x - \pi) \leq 1
\][/tex]
2. Transform the Sine Function:
Since the function inside the sine, [tex]\(x - \pi\)[/tex], is just a horizontal shift of the basic sine wave, it does not affect the amplitude or range of the sine wave. Therefore, the above inequality holds regardless of the shift:
[tex]\[
-1 \leq \sin(x - \pi) \leq 1
\][/tex]
3. Apply the Transformation to the Sine Output:
Next, the function scales the output of [tex]\(\sin(x - \pi)\)[/tex] by 2:
[tex]\[
2 \sin(x - \pi)
\][/tex]
Multiplying the entire range [tex]\([-1, 1]\)[/tex] by 2 changes the inequality to:
[tex]\[
-2 \leq 2 \sin(x - \pi) \leq 2
\][/tex]
4. Shift the Resulting Range:
The final function shifts the result vertically by adding 1:
[tex]\[
1 + 2 \sin(x - \pi)
\][/tex]
Adding 1 to each part of the inequality [tex]\(-2 \leq 2 \sin(x - \pi) \leq 2\)[/tex] gives:
[tex]\[
-1 \leq 1 + 2 \sin(x - \pi) \leq 3
\][/tex]
Thus, the range of the function [tex]\( y = 1 + 2 \sin(x - \pi) \)[/tex] is from [tex]\(-1\)[/tex] to [tex]\(3\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{-1 \text{ to } 3} \][/tex]
