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The domain of a function is essentially the collection of all possible x-values that will work in the function without causing any issues. In simpler terms, it is the set of inputs that lead to valid outputs. For a function with a square root, like \(f(x) = \sqrt{16 - 2x^2}\), the expression inside the square root (known as the radicand) must be zero or positive. Negative values under a square root would result in imaginary numbers, which are not defined in the set of real numbers. To find the domain, solve the inequality \(16 - 2x^2 \geq 0\), simplifying to \(x^2 \leq 8\). Taking the square root of both sides will give \(-\sqrt{8} \leq x \leq \sqrt{8}\). Finally, simplify \(\sqrt{8}\) to \(2\sqrt{2}\). Thus, the domain of \(f(x)\) is all x-values between \(-2\sqrt{2}\) and \(2\sqrt{2}\), inclusive. These values ensure that the function outputs real numbers.

The range of a function represents all possible y-values or outputs that the function can achieve. Understanding the range requires examining how the function behaves across its domain. For \( f(x) = \sqrt{16 - 2x^2} \), the range is determined by the maximum and minimum values of the function.Since square root functions produce non-negative results, the lowest y-value is zero. This occurs when the radicand is zero. For our function, this happens at the endpoints of the domain where \(x\) is \(\pm 2\sqrt{2}\), resulting in \(f(x) = 0\).The maximum value occurs when the radicand maximum is achieved, specifically when \(x = 0\). Substitute \(x = 0\) in the function to get \(f(x) = \sqrt{16} = 4\). Therefore, the range of the function is from 0 to 4, noted as \[0, 4\]. These are the outputs you're guaranteed to get from plugging in values from the domain.

Inequalities are expressions that show the relationship between two values where they are not equal. In the context of functions, inequalities help determine the domain because they tell us which x-values will result in real outputs. For the function \(f(x) = \sqrt{16 - 2x^2}\), the inequality \(16 - 2x^2 \geq 0\) ensures the square root yields real numbers. Solving this requires techniques such as isolating \(x\) through algebraic manipulation. Start by dividing through by 2: \(8 - x^2 \geq 0\), which transforms to \(x^2 \leq 8\). Then, taking the square root of both sides gives \(-\sqrt{8} \leq x \leq \sqrt{8}\). This process ensures every step adheres to the properties of inequality, wrapping up with a domain of \([-2\sqrt{2}, 2\sqrt{2}]\).Understanding inequalities is vital, not just for domains, but whenever you're tasked with finding conditions under which a statement holds true.