What is the range of [tex]$y=-5 \sin (x)$[/tex]?
A. all real numbers
B. [tex]$-5 \leq y \leq 5$[/tex]
C. all real numbers [tex]$-\frac{5}{2} \leq y \leq \frac{5}{2}$[/tex]
D. all real numbers [tex]$-1 \leq y \leq 1$[/tex]
E. all real numbers [tex]$-\frac{1}{5} \leq y \leq \frac{1}{5}$[/tex]
Answer (2)
The range of the function y = − 5 sin ( x ) is all values between -5 and 5, inclusive. Thus, the correct answer is option B: − 5 ≤ y ≤ 5 .
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The range of sin ( x ) is − 1 ≤ sin ( x ) ≤ 1 .
Multiplying by -5 reverses the inequality signs: 5 ≥ − 5 sin ( x ) ≥ − 5 .
Rewriting the inequality gives − 5 ≤ − 5 sin ( x ) ≤ 5 .
Therefore, the range of y = − 5 sin ( x ) is − 5 ≤ y ≤ 5 .
Explanation
Understanding the Problem We are asked to find the range of the function y = − 5 sin ( x ) . The range of a function is the set of all possible output values (y-values) that the function can produce.
Range of Sine Function We know that the sine function, sin ( x ) , oscillates between -1 and 1, inclusive. That is, for any value of x , we have − 1 ≤ sin ( x ) ≤ 1 .
Multiplying by -5 Now, we need to find the range of − 5 sin ( x ) . To do this, we multiply the inequality − 1 ≤ sin ( x ) ≤ 1 by -5. Remember that when we multiply an inequality by a negative number, we must reverse the direction of the inequality signs. So, we get: − 5 ( − 1 ) ≥ − 5 sin ( x ) ≥ − 5 ( 1 ) 5 ≥ − 5 sin ( x ) ≥ − 5
Finding the Range of y We can rewrite this inequality as: − 5 ≤ − 5 sin ( x ) ≤ 5 Since y = − 5 sin ( x ) , we can substitute y into the inequality: − 5 ≤ y ≤ 5 Thus, the range of the function y = − 5 sin ( x ) is all real numbers between -5 and 5, inclusive.
Final Answer The range of y = − 5 sin ( x ) is − 5 ≤ y ≤ 5 .
Examples
Understanding the range of trigonometric functions like y = − 5 sin ( x ) is crucial in many real-world applications. For example, consider an electrical circuit where the voltage varies sinusoidally with time. If the voltage is given by V ( t ) = − 5 sin ( t ) , then knowing the range tells us that the voltage will always be between -5 volts and 5 volts. This information is essential for designing the circuit and ensuring that components are not subjected to voltages beyond their rated limits. Similarly, in mechanical systems involving oscillations, such as a pendulum or a spring, the displacement can often be modeled using sinusoidal functions. The range of these functions helps determine the maximum displacement or amplitude of the oscillation.
