The range of the function f(x) = logₑ √(4 − x²), is:
2019
The range of the function f(x) = logₑ √(4 − x²), is:
- A.
(ln 2, ∞)
- B.
(−∞, ∞)
- C.
(−∞, ln 2)
- D.
(0, ∞)
Attempted by 24 students.
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Correct answer: C
Rewrite the function:
f(x) = ln √(4 − x²) = (1/2) ln(4 − x²).
Domain: Require the logarithm argument to be positive, so √(4 − x²) > 0 ⇒ 4 − x² > 0 ⇒ −2 < x < 2.
Range analysis: For x in (−2, 2), 4 − x² lies in (0, 4], so √(4 − x²) lies in (0, 2]. Taking natural logarithm gives ln(√(4 − x²)) ∈ (−∞, ln 2]. Note that ln 2 is attained at x = 0 because √(4 − 0²) = 2.
Final answer: The range of f is (−∞, ln 2].