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Understanding the concept of domain and range is crucial when dealing with functions and their inverses. The **domain** of a function describes all the possible input values (often represented on the x-axis), while the **range** refers to all possible output values (often represented on the y-axis).

For the function \(f(x) = \frac{1}{2} \sqrt{4-x^2}\), the domain on the restricted interval is \([0, 2]\). This means that only values of \(x\) within this interval can be input into the function. Within this domain, the **range** is determined by evaluating the function at the boundaries: \(f(0) = 1\) and \(f(2) = 0\), showing values decrease from 1 to 0.

When finding the inverse, the domain and range of \(f\) and \(f^{-1}\) swap. Thus, \(f^{-1}\) has a domain of \([0, 1]\) and range \([0, 2]\). Understanding this interchange helps in analyzing how the inverse maps the output back to the input values of \(f\).

A function must be **one-to-one** for it to have an inverse. This means that each input must correspond to exactly one output, and vice versa. A simple test for determining if a function is one-to-one is the *Horizontal Line Test*. If no horizontal line intersects the graph of the function at more than one point, the function is one-to-one.

In this exercise, the function \( f(x) = \frac{1}{2} \sqrt{4-x^2} \) over the restricted domain \([0, 2]\) passes the Horizontal Line Test, making it one-to-one. This distinction is vital because it ensures that the function has an inverse.

The need for the function to be one-to-one on a restricted domain, rather than the full domain, arises because the full domain might cause overlaps in outputs for different inputs, preventing a unique inverse function.

Graphing functions and their inverses can provide visual clarity on the relationship between them. When you graph a function and its inverse on the same axes, the graphs should be reflections across the line \(y = x\).

To sketch \(f(x) = \frac{1}{2} \sqrt{4-x^2}\) on \([0, 2]\), we get the upper half of a semicircle from \((0,1)\) to \((2,0)\). This represents output values of the function as x-moves from 0 to 2. For the inverse, \( f^{-1}(y) = 2\sqrt{1-y^2} \), draw a reflected semicircle on \([0, 1]\). This graph flips over the line \(y = x\), showing input values of \(f\) being mapped back from output values y-to-x.

Understanding these reflections clarify how domain and range switch roles and graphically support why an inverse function mirrors the original across the line \(y = x\). Providing a visual representation can simplify complex function behavior, helping one grasp inverse relationships more intuitively.