To determine the range of the function [tex]\(y = -3 \sin (x) - 4\)[/tex], we need to analyze the behavior of the sine function and how it affects the expression as a whole.
1. Understand the basic sine function:
The sine function, [tex]\(\sin(x)\)[/tex], oscillates between -1 and 1 for all real numbers [tex]\(x\)[/tex]. This means that the range of [tex]\(\sin(x)\)[/tex] is:
[tex]\[
-1 \leq \sin(x) \leq 1
\][/tex]
2. Transform the sine function:
The given function is [tex]\(y = -3 \sin(x) - 4\)[/tex], which involves scaling and shifting the sine function.
- The multiplication by -3 scales the sine function by 3 and flips it vertically. So, the range of [tex]\(-3 \sin(x)\)[/tex] becomes:
[tex]\[
-3 \leq -3 \sin(x) \leq 3
\][/tex]
- Next, subtracting 4 shifts the entire range down by 4 units. Thus, the range of [tex]\(-3 \sin(x) - 4\)[/tex] is:
[tex]\[
(-3 - 4) \leq -3 \sin(x) - 4 \leq (3 - 4)
\][/tex]
3. Simplify the range:
Simplify the bounds calculated above:
[tex]\[
-7 \leq -3 \sin(x) - 4 \leq -1
\][/tex]
Therefore, the range of the function [tex]\(y = -3 \sin (x) - 4\)[/tex] is:
[tex]\[
-7 \leq y \leq -1
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{all real numbers } -7 \leq y \leq -1
\][/tex]
