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Chapter 1: Problem 35

Determine the domain of the function represented by the given equation. $$f(x)=\sqrt{4-x^{2}}$$

Short Answer

Expert verified
The domain of the function \(f(x) = \sqrt{4 - x^{2}}\) is \([-2, 2]\).

Step by step solution

01

Set the expression under the root to greater than or equal to zero

Solve the inequality \(4 - x^{2} \geq 0\), this means that \(4 - x^{2}\) must be either positive or zero. Because if it is less than zero then it won't be a real number and won't be part of the domain.
02

Solve the inequality

Rearrange to have the x terms on one side, resulting in \( -x^{2} \geq -4\). It is important to remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign changes. Therefore, multiply through by -1 to get \(x^{2} \leq 4\). This form gives a clear idea of how x values are limited by 4.
03

Determine the range of x

Now, we have \(x^{2} \leq 4\). The solutions to this inequality in real numbers involves absolute values, as squaring eliminates any negative sign. This leads to \( -2 \leq x \leq 2\). This is the solution of inequality.

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