Graph the piecewise-defined function to determine whether it...

Graph the piecewise-defined function to determine whether it is a one-to-one function. If it is a one-to-one function, find its inverse.
$$f(x)=\left\{\begin{array}{ll}
-x & x<-2 \\
\sqrt{4-x^{2}} & -2 \leq x \leq 0 \\
-\frac{1}{x} & x>0
\end{array}\right.$$

Key Concepts

Inverse Function

The inverse function of a one-to-one function reverses the roles of the input and output, effectively swapping the domain and range. To find an inverse, one typically solves the equation y = f(x) for x in terms of y and then interchanges the variables. The resulting function, if it exists, undoes the action of the original function.

Domain and Range Considerations

When dealing with functions, especially piecewise-defined ones, it is essential to consider the restrictions on the domain and the corresponding range for each piece. These considerations are crucial when determining the overall behavior of the function and are particularly important when finding inverses, as the inverse function's domain is the original function's range and vice versa.

Horizontal Line Test

The horizontal line test is a visual method for determining whether a function is one-to-one. If every horizontal line drawn through the graph intersects the function at most once, the function is one-to-one. This test helps ascertain the invertibility of a function.

One-to-One Function

A one-to-one function is a function in which every element of the range is paired with exactly one unique element of the domain. This means no horizontal line intersects the graph of the function more than once, ensuring that the function has a unique inverse if one exists.

Piecewise-Defined Function

A piecewise-defined function is one that is described by different expressions over specific intervals of the domain. Each piece is defined by its own rule that applies only within a designated subdomain, and understanding how these pieces interact is crucial for analyzing the overall function's properties.

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