The domain of the function \(f(x) = \log\,(x^2-4)\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \((-\infty, -2) \cup (2, \infty)\)
The logarithm is defined only for positive numbers, so
\(x^2 - 4 \gt 0\)
\((x - 2)(x + 2) \gt 0\)
A product of two factors is positive when both factors have the same sign:
\(x \lt -2\) or \(x \gt 2\)
So the domain is \((-\infty, -2) \cup (2, \infty)\).
Why the other options are incorrect
Option (1) \([2, \infty)\) — Incorrect. At x = 2, \(x^2 - 4 = 0\) and \(\log 0\) is not defined; the values \(x \lt -2\) are also missing.
Option (3) \((2, \infty)\) — Incorrect. It leaves out \(x \lt -2\), where \(x^2 - 4\) is also positive (x = −3 gives 5).
Option (4) \((-\infty, -2] \cup (-2, -1)\) — Incorrect. It contains x = −2 (where \(x^2 - 4 = 0\)) and x = −1.5 (where \(x^2 - 4 = -1.75\)), and it misses every \(x \gt 2\).