Understanding the Domain of the Function f(x) √(sinx) - 1

In this comprehensive guide, we will explore how to determine the domain of the function f(x) √(sinx) - 1. This function involves a square root, which requires a careful analysis of its domain to ensure that the expression under the square root is non-negative. By the end of this article, you will have a clear understanding of how to find and understand the domain of such functions.

Introduction to the Function

The function in question is f(x) √(sinx) - 1. To understand its domain, we need to consider the expression under the square root, which is sinx - 1. The square root function is only defined for non-negative values.

Square Root Function Properties

The square root function, denoted as √x, is defined for all x ≥ 0. Therefore, for f(x) √(sinx - 1) to be well-defined, the expression inside the square root, sinx - 1, must be non-negative:

sinx - 1 ≥ 0

Solving the Inequality

To solve this inequality, let's isolate sinx:

sinx ≥ 1

Understanding the Range of sinx

The sine function, sinx, oscillates between -1 and 1. The inequality sinx ≥ 1 is only satisfied when sinx equals its maximum value of 1. This occurs at specific points:

This means that for the expression under the square root to be non-negative, sinx must be equal to 1. Therefore, the domain of the function f(x) √(sinx - 1) is:

Domain of the Function

The domain of the function is:

x π/2 2πn, where n is any integer.

Conclusion

In summary, the domain of the function f(x) √(sinx) - 1 is determined by the requirement that the expression under the square root, sinx - 1, be non-negative. This results in the domain being the set of x values for which sinx 1. Thus, the domain is:

x π/2 2πn, where n is any integer.

Frequently Asked Questions

1. Can the square root of a negative number be defined?

No, the square root of a negative number is not defined within the real number system. It is defined only for non-negative numbers.

2. Why does sinx need to be equal to 1?

For the expression under the square root, sinx - 1, to be non-negative, sinx must be greater than or equal to 1. Since the maximum value of sinx is 1, this is the only value that satisfies the condition.

3. What happens if we remove the -1 from the function?

If we remove the -1 from the function, we get f(x) √(sinx). The domain of this function is all x for which sinx is non-negative, i.e., sinx ≥ 0. This occurs for:

x ∈ [0 2πn, π 2πn], where n is any integer.