Question 4

The domain of the function f(x) = (√(x + 2))/(|x| - 1) is x ≥ -2, x ≠ -1, x ≠ 1.

To determine the domain of the function, we must ensure that the expression under the square root is non-negative and the denominator is not equal to zero. The conditions for the function f(x) require that x + 2 ≥ 0 (which leads to x ≥ -2) and |x| - 1 ≠ 0 (which leads to x ≠ -1 and x ≠ 1).

A) x ≥ -2

This option correctly identifies the requirement for the square root to be defined since x + 2 must be greater than or equal to zero. However, it fails to address the restrictions imposed by the denominator, which must also not equal zero.

B) x > -2, x ≠ 1

While this option correctly states that x must be greater than -2, it neglects the necessary condition that x cannot equal -1. Therefore, it is incomplete and does not correctly define the domain of the function.

C) x ≥ -2, x ≠ -1

This option captures the requirement for the square root to be defined with x + 2 ≥ 0 and correctly excludes -1 from the domain. However, it fails to exclude 1, where the denominator would also be zero, making this option insufficient.

D) x ≥ -2, x ≠ -1, x ≠ 1

This option accurately reflects all the conditions necessary for the function to be defined. It ensures that the square root is non-negative and that the denominator does not equal zero by excluding both -1 and 1, making it the correct answer.

E) x > -2, x ≠ -1, x ≠ 1

This option correctly states that x must be greater than -2 and excludes both -1 and 1 but is not precise on the inclusion of -2 itself. Thus, it is incomplete and does not capture the full domain.

Conclusion

Option D is definitively correct as it encompasses all necessary conditions for the function's domain, ensuring the square root is defined and the denominator is not zero. The other options fail by either missing necessary exclusions or not accounting for the complete range of x values that maintain the function's validity.