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Understanding the domain of a function is a fundamental concept in mathematics. It refers to all the possible input values (often represented as x-values) that you can use in the function without leading to any undefined situations, like division by zero or square roots of negative numbers. In the function we consider, \(f(x) = \sqrt{4 - x^2}\), finding the domain involves determining the values for which the expression inside the square root remains non-negative.

Here's how to identify the domain for \(f(x) = \sqrt{4 - x^2}\):

• The expression under the square root, \(4 - x^2\), must be greater than or equal to zero.
• Thus, solve the inequality \(4 - x^2 \geq 0\).
• Rearranging terms, you get \(x^2 \leq 4\).
• Taking the square root gives you the range of x-values as \(-2 \leq x \leq 2\).
• This means the domain is \([-2, 2]\).

Always remember, when you're dealing with square roots, ensuring the values remain non-negative is key. Checking the domain carefully is essential to plotting the correct range on your graph.

The range of a function refers to the set of all possible outputs (y-values) you can get by plugging in the domain values into the function. For the given function \(f(x) = \sqrt{4 - x^2}\), determining the range involves looking at all the values \(f(x)\) can possibly take.

Here's a breakdown on how to find the range:

• The function \(f(x)\) consists of a square root, which always results in a non-negative value.
• When \(x\) is 0, \(f(x) = \sqrt{4 - 0^2} = \sqrt{4} = 2\). This gives the maximum value of the function.
• As \(x\) approaches the edges of the domain (i.e., \(x = -2\) or \(x = 2\)), \(f(x)\) approaches 0, the minimum value.
• Thus, the range of \(f(x)\) spans from 0 to 2, inclusive.

Understanding the range helps in fully grasping the potential behavior of a function across its domain. In graphical terms, this tells you the vertical extent of the graph of \(f(x) = \sqrt{4 - x^2}\).

Inequalities in calculus often involve solving for conditions under which a particular expression remains valid or true. They are crucial when determining domains and are used frequently in analyzing functions. In our exercise, we use inequalities to find permissible values of \(x\) such that the function \(f(x) = \sqrt{4 - x^2}\) remains defined.

Here's a quick guide to dealing with inequalities like \(4 - x^2 \geq 0\):

• Start by isolating the inequality condition, such as \(4 - x^2 \geq 0\).
• Rearrange to the simpler form: \(x^2 \leq 4\).
• Take the square root of both sides, remembering that taking square roots of inequalities will inversely flip the inequality sign for negative solutions. In our case, \(-2 \leq x \leq 2\).
• The solution gives the allowable values of \(x\), ensuring that the expression under any square root is non-negative.

Understanding these steps is essential for grasping the broader aspect of solving equations and inequalities within calculus, especially when dealing with functions involving square roots or other constraints.