Determine the Domain of the Function f(x) √[√x √sinx] √cosx

When analyzing the function f(x) √[√x √sinx] √cosx, it is crucial to understand the domain in which the function is defined. The domain is the set of all input values (x) for which the function produces real number outputs. Here, we will break down the steps to determine the domain of this particular function.

Step 1: Simplify the Function

Let's simplify the function to make it more readable and easier to analyze. We will strip away the unnecessary parentheses and use standard math symbols.

Original function: f(x) √[√x √sinx] √cosx

Simplified function: f(x) √{√x √sinx} √cosx

Step 2: Analyze Trigonometric Constraints

First, let's analyze the constraints from the trigonometric functions, specifically cos x and sin x.

Condition 1: For √cosx to be real, cos x must be non-negative.

cos x 0 occurs in the first and fourth quadrants. The general intervals for cos x 0 are: [2nπ, 2nπ π/2].

Condition 2: For √sinx to be real, sin x must be non-negative.

sin x 0 occurs in the first and second quadrants. The general intervals for sin x 0 are: [2nπ, 2nπ π].

To satisfy both conditions simultaneously, x must be in the first quadrant. Therefore, the constraints combine to give the interval: [2nπ, 2nπ π/2] where n is an integer.

Step 3: Consider the Square Root of x

Next, we consider the constraint from √x.

For √x to be defined, x must be non-negative: x ≥ 0.

Merging this with the previous interval, we obtain: [2nπ, 2nπ π/2]. However, we need to restrict this interval further using the specific value of x 0.

Since the interval [2nπ, 2nπ π/2] already ensures that x is positive, we can simplify the domain to: [2nπ, 2nπ π/2] for even integers n, to match the requirement that the square root of a number is defined for positive numbers only.

Final Domain

Combining all the constraints, the domain of the function f(x) √[√x √sinx] √cosx is:

[2kπ, 2kπ π/2] where k is any positive integer. This interval ensures that all parts of the function are well-defined.

Note: The requirement for k to be a positive integer ensures that we only consider the first quadrant where both sine and cosine are positive, and x 0.

Conclusion

Understanding the domain of a function is crucial for its proper use and interpretation. By carefully analyzing the constraints from each component of the function, we can determine the valid range of x values that ensure the function f(x) √[√x √sinx] √cosx is defined in the real number system.