need help with this question

[Teen]

the range of the function f;[0,/3)-->R,f(x)=3*modulus[sin(2x)-1]+2 is

The question is from 2007 exam 2...Multiple choice . How do u it without technology 

Thanks in advance 

[BubbleWrapMan]

sin(2x) ≤ 1

So, sin(2x) - 1 ≤ 0, which means that |sin(2x) - 1| = 1 - sin(2x) (if it's 0 it stays zero, if it's negative it becomes positive)

So the function is really 3(1-sin(2x))+2 = 5 - 3sin(2x)

Now, note the period of this function is . Considering it always takes a sine graph a quarter of a period to reach its maximum or minimum from its inflection point (that is, the point where sin(x) is normally 0), we can say that it will have reached its minimum of 2, since , and also it will not have gone above 5 since , and it takes half a period to return to its point of inflection. Hence the range is [2, 5].