What is the domain of the function f(x) = (√(x + 2))/(|x| - 1) ?

Rationale

x ≥ -2, x ≠ -1, x ≠ 1

To find the domain of the function ( f(x) = frac{sqrt{x + 2}}{|x| - 1} ), we consider two restrictions. First, the square root requires ( x + 2 > 0 ), so ( x > -2 ). Second, the denominator cannot be zero. Since ( |x| - 1 = 0 ) when ( x = 1 ) or ( x = -1 ), these values must be excluded. Therefore, the domain is all real numbers greater than or equal to -2, except -1 and 1.

A (x ≥ -2) YOUR ANSWER

This includes values where the denominator becomes zero (x = -1 and x = 1). Since division by zero is undefined, this option is incomplete and incorrect.

B (x > -2, x ≠ 1) YOUR ANSWER

This incorrectly excludes x = -2, which is valid because it makes the square root equal to zero. It also fails to exclude x = -1, where the denominator is zero. Hence, this is incorrect.

C (x ≥ -2, x ≠ -1) YOUR ANSWER

This correctly includes the square root condition and excludes x = -1, but it does not exclude x = 1, which also makes the denominator zero. Therefore, it is incomplete.

D (x ≥ -2, x ≠ -1, x ≠ 1) CORRECT

This correctly applies both conditions: it includes all values where the square root is defined and excludes all values that make the denominator zero. This is the correct answer.

E (x > -2, x ≠ -1, x ≠ 1) YOUR ANSWER

This incorrectly excludes x = -2, even though it is allowed. The square root of zero is defined, so this restriction is unnecessary, making the option incorrect.

Conclusion

The domain is all real numbers greater than or equal to -2, excluding -1 and 1. Option D correctly represents all valid inputs while excluding values that make the function undefined

What is the domain of the function f(x) = (√(x + 2))/(|x| - 1) ?

x ≥ -2, x ≠ -1, x ≠ 1

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