Question

Let \( f(x) = \sqrt{4 - x^2} \), \( g(x) = \sqrt{x^2 - 1} \). Then the domain of the function \( h(x) = f(x) + g(x) \) is equal to:

• A. \( (-\infty, -1] \cup [1, \infty) \)
• B. \( (-\infty, -2] \cup [2, \infty) \)
• C. \( [-2, -1] \cup [1, 2] \)
• D. \( [-2, 1] \cup [1, 2] \)
• E. \( [1, 2] \)

Hint:

To find the domain of a composite function, determine the common domain for all functions involved.

Answer 1

The Correct Option is D

The domain of \( f(x) = \sqrt{4 - x^2} \) is \( -2 \leq x \leq 2 \), and the domain of \( g(x) = \sqrt{x^2 - 1} \) is \( x \leq -1 \) or \( x \geq 1 \). For the function \( h(x) = f(x) + g(x) \) to be defined, the domain must satisfy both conditions: Thus, the domain of \( h(x) \) is \( [-2, 1] \cup [1, 2] \). 
Thus, the correct answer is (D).