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Chapter 1: Problem 36

Determine the domain of the function represented by the given equation. $$f(x)=\sqrt{12-x^{2}}$$

Short Answer

Expert verified
The domain of the function f(x) = \( \sqrt{12-x^{2}} \) is \( -2\sqrt{3} \leq x \leq 2\sqrt{3} \).

Step by step solution

01

Set up the inequality

Write an inequality that expresses that the radicand (the expression inside the square root) must be equal to or greater than zero (0) in order for the square root to output a real number. This gives you \( 12 - x^2 \geq 0 \).
02

Solve the inequality

Avoid dealing with negative x by flipping the sign of the x term and the inequality. This gives you \( x^2 - 12 \leq 0 \). Then move 12 to the other side to find the x-value: \( x^2 \leq 12 \). Take the square root of both sides to solve for x, this gives you \( -\sqrt{12} \leq x \leq \sqrt{12} \). Also, the square root of 12 can be simplified to 2 times the square root of 3, so the range can be given as \( -2\sqrt{3} \leq x \leq 2\sqrt{3} \).
03

State the domain

The previous step has obtained the domain of the function. This domain includes all x-values that can be used in the function to get a real output. So the domain of \( f(x)=\sqrt{12-x^{2}} \) is \( -2\sqrt{3} \leq x \leq 2\sqrt{3} \).

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