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The domain of composed functions is crucial for understanding where the function is defined. When you have a composition of two functions, say \((f \circ g)(x) = f(g(x))\), you need to look at where both functions are valid. For instance, if the inner function \g(x)\ has a domain restriction, those restrictions must also hold for \f(g(x))\. Then you must also check if the resulting expression from \f(g(x))\ has any new domain limitations. Consider the example of \f(x) = \sqrt{x + 4}\ and \g(x) = -\frac{2}{x}\. For \f(g(x))\, you need to ensure the expression under the square root is non-negative and that \g(x)\ itself is defined. Always examine the composition carefully: find where \g(x)\ is valid and then see where \f\ applied to \g(x)\ remains valid.

Function notation is a concise way to express the idea of functions and their compositions. For example, \(f(x) = \sqrt{x + 4}\) indicates that function \(f\) takes an input \(x\) and maps it to \sqrt{x + 4}\. Similarly, \(g(x) = -\frac{2}{x}\) takes an input \(x\) and maps it to \ -\frac{2}{x}\. When you see notation like \((f \circ g)(x)\), read it as 'f of g of x' or 'f composed with g at x'. This means you first apply \g(x)\ and then apply \f\ to the result of \g(x)\. Understanding this notation helps to visualize the step-by-step process needed to find the resulting values and domains.

Inequalities play a significant role in determining the domains of composed functions. If you have a function like \(f(x) = \sqrt{x + 4}\), the argument, \(x + 4,\) must be non-negative because square roots of negative numbers are undefined in the real number system. This gives the inequality \x + 4 \ge 0\, simplifying to \x \ge -4\. When composing functions, like \(f(g(x))\), you'll need to solve inequalities to ensure that the expression inside the square root is always non-negative. For example, in the exercise, solving \( -\frac{2}{x} + 4 \ge 0\) led to finding the domain. Such steps guarantee the composed function is valid over specified intervals of \x\.

Square roots, by definition, only produce non-negative outputs. The expression \sqrt{x}\ is only defined for \x\ values that are zero or positive. Whenever you read \sqrt{}\, remember it implies a domain restriction: the input must be non-negative. Additionally, when dealing with square roots in composed functions, like \(f(x) = \sqrt{x + 4}\), ensure the entire expression inside the square root, \(x + 4\), is non-negative. Always set up an inequality reflecting this constraint, solve it, and then determine the interval where the square root function is defined. This understanding helps to avoid mistakes when determining valid inputs for composed functions.