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QUESTION 33
What is the range of the function
given that ?

Ask by Paul Willis. in Papua New Guinea
Jan 24,2025

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The range of when is all real numbers from –4 to 4.

Solución

The function given is with the domain .
  1. Understanding the Sine Function:
    • The basic sine function, , oscillates between -1 and 1. That is, has a range of .
  2. Scaling the Sine Function:
    • The function scales the sine function vertically by a factor of 4. This means that every output value of is multiplied by 4.
    • Therefore, the range of becomes .
  3. Domain Consideration:
    • Since is restricted to the interval , we examine the sine function over one full period. Within this interval, indeed achieves all values between -1 and 1.
Conclusion:
The range of the function over the interval is .
Answer: Its range is all real numbers from –4 up to 4.

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Beyond the Answer

The function oscillates between its minimum and maximum values based on the sine function. Since achieves values from to , when multiplied by , the range of becomes . Therefore, as varies from to , will take on every value between these two bounds.
To visualize this, imagine plotting the function: as you move along the x-axis from to , you’ll see the graph peaking at and dipping down to like a rollercoaster of sine waves. It’s an exciting ride! Just remember, no matter where you are in this interval, will always stay within that thrilling range from to .

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