It should lie between -(13)^(1/2) to +(13)^(1/2). any expression of the for of asinx+bcosx lies in the range of-(a^2+b^2)^(1/2) to +(a^2+b^2)^(1/2)

Range of 3|sinx| - 2|cosx|
Understanding the given expression is crucial to determine its range. Let's break it down step-by-step:

Breaking down the expression:
- The expression is 3|sinx| - 2|cosx|
- The modulus function ensures that the value inside it is always non-negative.
- For sinx, the range is [-1, 1] and for cosx, the range is also [-1, 1].

Determining the range:
- When sinx and cosx are positive, their absolute values are equal to the original values.
- When sinx is negative, |sinx| becomes -sinx, and when cosx is negative, |cosx| becomes -cosx.
- Thus, the given expression becomes 3sinx - 2cosx for sinx > 0, and 3sinx + 2cosx for cosx > 0.

Range for sinx > 0:
- As sinx varies from 0 to 1, the expression ranges from 3(0) - 2(1) = -2 to 3(1) - 2(0) = 3.
- Therefore, for sinx > 0, the range is [-2, 3].

Range for cosx > 0:
- As cosx varies from 0 to 1, the expression ranges from 3(1) - 2(0) = 3 to 3(0) + 2(1) = 2.
- Therefore, for cosx > 0, the range is [2, 3].

Combining the ranges:
- By considering both cases, the overall range for the expression 3|sinx| - 2|cosx| is [-2, 3] ∪ [2, 3].
- This means the range includes all values between -2 and 3, as well as values between 2 and 3.
In conclusion, the range of the expression 3|sinx| - 2|cosx| is [-2, 3] ∪ [2, 3], encompassing a range of values that satisfy the given conditions.