The unit circle is not a function. If we restrict ourselves...

The unit circle is not a function. If we restrict ourselves to the semicircle that lies in quadrants I and II, the graph represents a function, but it is not a one-to-one function. If we further restrict ourselves to the quarter circle lying in quadrant $\mathrm{I},$ the graph does represent a one-to-one function. Determine the equations of both the one-to-one function and its inverse. State the domain and range of both.

Key Concepts

Graph Restrictions

Graphical restrictions involve limiting the domain (and sometimes the range) of a relation to obtain a function that possesses desired properties, such as being one-to-one. For example, restricting the unit circle to only a quarter (in quadrant I) results in a one-to-one function that can then have an inverse function.

Domain and Range

The domain of a function is the set of all allowable input values, and the range is the set of all resulting output values. When considering inverse functions, the domain and range are interchanged; the domain of the original function becomes the range of its inverse and vice versa.

Inverse Function

An inverse function reverses the input-output roles of the original function. When a function is one-to-one, its inverse exists and for every output of the function, the inverse function returns the corresponding original input.

One-to-One Function

A one-to-one function, or injective function, is a function in which no two different inputs produce the same output. This property is crucial for the existence of an inverse function, since each output must be uniquely linked to one input to reverse the mapping.

Function

A function is a relation between a set of inputs and outputs where every input is paired with exactly one output. In the context of the unit circle, the full circle does not represent a function because there exist x-values with two corresponding y-values, violating the definition of a function.

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