Sets & FunctionsMedium

Domain of a composite function

Determine the domain of f(x) = ln(x² − 4) + √(9 − x²).

Given data

f(x) = ln(x² − 4) + √(9 − x²)

Steps

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What is the existence condition for the logarithm ln(x² − 4)?
Write your working — the numerical result must appear in the text (insieme)
What is the existence condition for the square root √(9 − x²)?
What is the intersection of the two intervals? (left endpoint)
What is the intersection of the two intervals? (right endpoint)

Full worked solution

What is the existence condition for the logarithm ln(x² − 4)?
$x^2 - 4 > 0$
$x^2 - 4 > 0 \rightarrow x^2 > 4 \rightarrow x < -2$ or $x > 2$ , i.e. $(-\infty, -2) \cup (2, +\infty)$ . The logarithm requires a strictly positive argument.
What is the existence condition for the square root √(9 − x²)?
$9 - x^2 \geq 0$
$9 - x^2 \ge 0 \rightarrow x^2 \le 9 \rightarrow -3 \le x \le 3$ , i.e. $[-3, 3]$ . The square root requires a non-negative radicand.
What is the intersection of the two intervals? (left endpoint)
$[-3, -2)$
Intersection on the left side: $[-3, -2)$ . The domain is the set of $x$ satisfying both conditions simultaneously.
What is the intersection of the two intervals? (right endpoint)
$(2, 3]$
Intersection on the right side: $(2, 3]$ . Full domain: $[-3, -2) \cup (2, 3]$ .

Result:Domain: [−3, −2) ∪ (2, 3].

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