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Understanding the domain and range of a function is crucial to determine whether two functions are inverses of each other. The domain of a function is the set of all possible input values, while the range is the set of all possible outputs.

For the function \( f(x) = -x^2 + 3 \) with \( x \geq 0 \), the domain is \([0, \infty)\). Since this function describes a downward-opening parabola, the maximum value is at the vertex, located at \( x = 0 \). Thus, the range is \((-\infty, 3]\).

The function \( g(x) = \sqrt{3 - x} \), with \( x \leq 3 \), has a domain of \((-\infty, 3]\). For square root functions like this, all output values are non-negative, resulting in the range \([0, \infty)\).

Both functions have reciprocally matching domains and ranges, required for inverse functions. This pairing is key for their inverse relationship.

The composition of functions helps verify the inverse relationship. To prove two functions are inverses, we demonstrate that composing them in both orders returns the input value. Specifically, if \( f \) and \( g \) are inverses, then \( f(g(x)) = x \) and \( g(f(x)) = x \) should both hold true.

For the given exercise, the function \( f(g(x)) = x \) for all \( x \leq 3 \) can be shown by substituting \( g(x) = \sqrt{3-x} \) into \( f(x) \). This simplifies to \(- (3 - x) + 3 = x\), confirming that \(f(g(x)) = x\).

Similarly, \( g(f(x)) = x \) for all \( x \geq 0 \) involves substituting \( f(x) = -x^2 + 3 \) into \( g(x) \), simplifying to \( \sqrt{x^2} = x \). Given that \( x \geq 0 \), \( \sqrt{x^2} = x \) is valid, proving \( g(f(x)) = x \).

Through successful composition in both orders, we conclude that \( f \) and \( g \) are inverses.

Graphing helps visualize the inverse relationship between functions. When graphed, inverse functions reflect over the line \( y = x \).

To sketch the graph of \( f(x) = -x^2 + 3 \), draw a half-parabola opening downwards, beginning at \( x = 0 \) and reaching a peak at \( (0, 3) \). This plot represents all non-negative \( x \) values.

For \( g(x) = \sqrt{3 - x} \), start at \( x = 3 \) and draw a curve that extends to the left, going upwards from zero. This curve matches all non-positive \( x \) values.

These graphs should mirror each other over the line \( y = x \), illustrating their inverse nature. Observing graph symmetries is a visually intuitive way to confirm that two functions are inverses.