Determining the Domain of the Function f(x) arcsin(√(x^2 - 1))

In order to determine the domain of f(x) arcsin(√(x^2 - 1)), we must examine the mathematical conditions imposed by both the square root function and the arcsine function. This article provides a step-by-step guide to finding this domain.

Understanding the Square Root Condition

The expression inside the square root, x^2 - 1, must be non-negative for the square root to be defined:

We start by setting up the inequality: x^2 - 1 ≥ 0 This simplifies to: x^2 ≥ 1 By taking the square root of both sides, we get: |x| ≥ 1, which implies two cases: x ≥ 1 x ≤ -1

Interpreting the Arcsin Condition

The output of the square root, √(x^2 - 1), must be within the range [-1, 1] for the arcsine function to be defined. Since the square root of a non-negative number is always non-negative, our condition simplifies to:

We set up the inequality: √(x^2 - 1) ≤ 1 Squaring both sides gives: x^2 - 1 ≤ 1 This simplifies to: x^2 ≤ 2 By taking the square root again, we get: |x| ≤ √2, which implies: -√2 ≤ x ≤ √2

Combining Conditions

To find the domain of f(x) arcsin(√(x^2 - 1)), we need to combine the two conditions:

From the square root condition: x ≤ -1 or x ≥ 1 From the arcsine condition: -√2 ≤ x ≤ √2

The domain that satisfies both conditions is the intersection of these intervals:

Final Domain

The valid intervals where both conditions are met are:

-√2 ≤ x ≤ -1 1 ≤ x ≤ √2

Thus, the domain of the function f(x) arcsin(√(x^2 - 1)) is:

[ -√2, -1 ] ∪ [ 1, √2 ]

Conclusion

By carefully examining the conditions for the square root and the arcsine functions, we have determined the domain of the given function. This process is essential for understanding the behavior and applicability of the function within specified intervals.