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.
$$f(x)=\left\{\begin{array}{ll}
\sqrt[3]{x} & x \leq-1 \\
x^{2}+2 x & -1<x \leq 1 \\
\sqrt{x}+2 & x>1
\end{array}\right.$$

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Key Concepts

Piecewise Functions

A piecewise function is one that is defined by different formulas over different parts of its domain. Understanding piecewise functions involves analyzing each segment separately, considering their respective domains, and then combining those analyses to study the overall behavior of the function.

One-to-One Functions

A function is one-to-one if each output value is produced by exactly one input value. This property ensures that no horizontal line touches the graph more than once—a concept known as the horizontal line test. Establishing one-to-one behavior is essential before attempting to determine an inverse function.

Graphing Functions

Graphing is an important analytical tool for understanding functions. By plotting the function, one can visually inspect behaviors such as changes in direction and verify if the function passes the horizontal line test, which is a practical approach to determine if a function is one-to-one.

Inverse Functions

The inverse of a function reverses the original mapping of inputs to outputs. For a function to have an inverse, it must be one-to-one, ensuring that each output corresponds uniquely to an input. In finding an inverse, one typically swaps the roles of the dependent and independent variables, and then solves for the new dependent variable.

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