To determine the range of the function [tex]\( y = -3 \sin(x) - 4 \)[/tex], we need to understand the behavior of the sine function and how it affects the given equation.
1. Understand the range of [tex]\( \sin(x) \)[/tex]:
The sine function, [tex]\(\sin(x)\)[/tex], oscillates between -1 and 1. Therefore, the range of [tex]\(\sin(x)\)[/tex] is [tex]\([-1, 1]\)[/tex].
2. Apply the transformation [tex]\( \sin(x) \)[/tex] to the given equation:
The equation we are working with is [tex]\( y = -3 \sin(x) - 4 \)[/tex].
3. Find the minimum value of [tex]\( y \)[/tex]:
The minimum value of [tex]\(\sin(x)\)[/tex] is -1. Substituting [tex]\(\sin(x) = -1\)[/tex] into the equation:
[tex]\[
y = -3(-1) - 4 = 3 - 4 = -1
\][/tex]
Therefore, the minimum value of [tex]\( y \)[/tex] is -1.
4. Find the maximum value of [tex]\( y \)[/tex]:
The maximum value of [tex]\(\sin(x)\)[/tex] is 1. Substituting [tex]\(\sin(x) = 1\)[/tex] into the equation:
[tex]\[
y = -3(1) - 4 = -3 - 4 = -7
\][/tex]
Therefore, the maximum value of [tex]\( y \)[/tex] is -7.
5. Determine the range:
Therefore, the range of [tex]\( y \)[/tex] is between -7 and -1. This means [tex]\( y \)[/tex] takes all real values within the interval [tex]\([-7, -1]\)[/tex].
Thus, the range of the function [tex]\( y = -3 \sin(x) - 4 \)[/tex] is all real numbers [tex]\(-7 \leq y \leq -1\)[/tex].
The correct choice is:
[tex]\[
\text{all real numbers } -7 \leq y \leq -1
\][/tex]
