Which interval gives the range of the function $y = 5 + 2 ext{cos}(3x)$?
A - HSC - SSCE Mathematics Advanced - Question 8 - 2020 - Paper 1
Question 8
Which interval gives the range of the function $y = 5 + 2 ext{cos}(3x)$?
A. [2, 8]
B. [3, 7]
C. [4, 6]
D. [5, 9]
Worked Solution & Example Answer:Which interval gives the range of the function $y = 5 + 2 ext{cos}(3x)$?
A - HSC - SSCE Mathematics Advanced - Question 8 - 2020 - Paper 1
Step 1
Identify the function
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Answer
The function given is y=5+2extcos(3x). This is a cosine function which oscillates between -1 and 1.
Step 2
Determine the range of the cosine function
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Answer
The range of extcos(3x) is between -1 and 1. Therefore, the oscillation of the function y=5+2extcos(3x) will be transformed by the coefficients.
Step 3
Calculate the maximum and minimum values
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Answer
The minimum value of the function occurs when extcos(3x)=−1, which gives:
ymin=5+2(−1)=5−2=3
The maximum value occurs when extcos(3x)=1, yielding:
ymax=5+2(1)=5+2=7
Step 4
Establish the range
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Answer
Thus, the range of the function is from 3 to 7, or in interval notation: [3, 7].
Step 5
Select the correct answer
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Answer
The correct interval that gives the range of the function is option B: [3, 7].
