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The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function \(f(x) = \sqrt{16 - 4x^2}\), the expression inside the square root, \(16 - 4x^2\), must be non-negative. This is because the square root of a negative number is not defined within the set of real numbers. To find the domain, we set up the inequality: \(16 - 4x^2 \geq 0\). Simplifying, we get \(4(4 - x^2) \geq 0\), which leads to \(4 - x^2 \geq 0\). Solving for \(x\), we find that \( -2 \leq x \leq 2\). Thus, the domain of the function is \[-2, 2\]\. This means that \(x\) can take any value from \(-2\) to \(2\), inclusive.

The concept of domain is crucial in understanding which values are valid inputs for a function. Without determining the domain, we might incorrectly assume the function is valid for all \(x\)-values, leading to errors in graphing and solving the function.

Intercepts are points where the graph of a function crosses the axes. There are two types of intercepts: x-intercepts and y-intercepts.

To find the x-intercepts of \(f(x) = \sqrt{16 - 4x^2}\), set the function equal to zero: \(\sqrt{16 - 4x^2} = 0\). Solving \(16 - 4x^2 = 0\) yields \(x = \pm 2\). Therefore, the x-intercepts are at points \((2, 0)\) and \((-2, 0)\). These are the points where the graph touches the x-axis.

To find the y-intercept, set \(x = 0\), and solve for \(f(0)\):\(f(0) = \sqrt{16} = 4\). Thus, the y-intercept is at \((0, 4)\). This is the point where the graph touches the y-axis.

Labeling these intercepts on the graph helps in sketching the function accurately, showing where it crosses the axes.

Inequalities are mathematical expressions involving the symbols \(<\), \(>\), \(≤\), and \(≥\). They are used to describe the range of values that satisfy a particular condition. In the context of graphing functions, inequalities help determine the domain and valid x-values.

For the function \(f(x) = \sqrt{16 - 4x^2}\), the inequality \(16 - 4x^2 \geq 0\) is set up to ensure the expression inside the square root is non-negative. Solving this inequality involves a few steps:

• First, recognize that \(16 - 4x^2\) is a quadratic expression.


• Factor it as \(4(4 - x^2) \geq 0\).


• Next, rewrite it as \(4 - x^2 \geq 0\).


• Finally, solve the inequality to find \(-2 \leq x \leq 2\).

This result means that x must be between \(-2\) and \(2\), inclusive, for the function to have real values. Understanding inequalities is essential for correctly determining the range of input values that maintain the function's validity.