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One-to-one functions are a special type of function where every input corresponds to exactly one unique output. This property makes each input value uniquely paired with one and only one output value. In other words, if you pick any two different inputs into the function, you will get two different outputs. This distinct pairing is crucial when dealing with inverse functions.

To determine if a function is one-to-one, you can use the "Horizontal Line Test." This involves drawing horizontal lines through the graph of the function. If any horizontal line crosses the graph more than once, the function is not one-to-one. However, if each horizontal line intersects the graph at most once, the function is one-to-one. For the function given, \(g(x) = x^2 + 1\) with \(x \geq 0\), we only need to consider non-negative values of \(x\), ensuring the function is monotonic (increasing in this case) and hence one-to-one.

Finding the inverse of a one-to-one function means solving for \(x\) in terms of \(y\). This involves reversing the role of inputs and outputs, ultimately leading to one unique inverse function that maps outputs back to inputs.

The concept of reflection across the line \(y=x\) is critical to understanding inverse functions. When two functions are mirror images of one another about the line \(y=x\), they are confirmed to be inverses of each other. This line, \(y=x\), acts as a mirror where everything on the graph flips across.

Think about this idea of reflection as flipping the coordinates of points on a graph. So, if a point \((a, b)\) is on the original function, the point \((b, a)\) will be on the inverse function. This interchange is depicted graphically by reflecting over the line \(y=x\).

In our example, if you graph \(g(x) = x^2 + 1\) and its inverse \(g^{-1}(x) = \sqrt{x - 1}\), you will observe that the graphs are indeed mirror images about the line \(y=x\). This visual confirmation is a powerful tool for understanding and verifying inverse functions.

Graphing functions is a fundamental skill in mathematics that helps in visualizing how functions behave and interact. By plotting the original function and its inverse, along with the line \(y=x\), we can observe their properties and ensure they align correctly as inverse functions.

For graphing, it's crucial to understand the behavior of each function first. The original function \(g(x) = x^2 + 1\) for \(x \geq 0\) represents a parabola shifted one unit upwards, with the vertex at \((0,1)\). The inverse, \(g^{-1}(x) = \sqrt{x - 1}\), starts at \((1,0)\) and moves to the right as \(x\) increases.

When these are plotted alongside the line \(y=x\), you should be able to see that \(g(x)\) and \(g^{-1}(x)\) are geometric reflections across \(y=x\). This graphical representation solidifies the concept of inverse functions and their relationship, providing a clear, visual check of your calculations.