Graph the piecewise-defined functions. State the domain and...

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.
$$G(x)=\left\{\begin{array}{ll}
0 & x=0 \\
-\frac{1}{x} & x \neq 0
\end{array}\right.$$

Key Concepts

Graphing Techniques for Functions

Graphing involves plotting the set of all points that satisfy the function equation. When graphing piecewise-defined functions, each segment must be plotted according to its specific rule over the corresponding interval, paying attention to open and closed endpoints, which represent exclusions or inclusions in the domain.

Interval Notation

Interval notation is a shorthand method for expressing subsets of the real numbers, particularly useful for indicating the domain and range of a function. It uses brackets and parentheses to denote whether endpoints are included or excluded.

Monotonicity Analysis

Monotonicity refers to the behavior of a function in terms of whether it is increasing, decreasing, or constant on intervals. Increasing means the function's output values rise as the input increases, decreasing means they fall, and constant means there is no change over those intervals.

Piecewise-Defined Functions

A piecewise-defined function is one that is defined by different expressions over different parts of its domain. This type of function requires checking which rule applies based on the specific input value, thus it may behave very differently on different intervals.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all output values the function can produce. Determining both involves considering restrictions such as division by zero or other constraints imposed by the function’s formula.

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