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What is the range of [tex]\( y = -3 \sin(x) - 4 \)[/tex]?

A. all real numbers [tex]\(-7 \leq y \leq 7\)[/tex]

B. all real numbers [tex]\(-7 \leq y \leq -1\)[/tex]

C. all real numbers [tex]\(-5 \leq y \leq 3\)[/tex]

D. all real numbers [tex]\(-1 \leq y \leq 1\)[/tex]

Sagot :

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]