Graphing Piecewise Defined Functions Sketch a graph of the...

Graphing Piecewise Defined Functions Sketch a graph of the piecewise defined function.
$$f(x)=\left\{\begin{array}{ll}{-1} & {\text { if } x<-1} \\ {1} & {\text { if }-1 \leq x \leq 1} \\ {-1} & {\text { if } x>1}\end{array}\right.$$

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Key Concepts

Discontinuity at Boundary Points

Discontinuities can occur at the points where the definition of the function changes. These areas, often at the endpoints of the intervals, need to be carefully analyzed to determine if the function is continuous or if there is a jump or removable discontinuity between the pieces.

Closed and Open Intervals

Closed intervals, denoted by brackets [ ], include the endpoints of the interval, while open intervals, denoted by parentheses ( ), exclude them. This distinction is critical when graphing piecewise functions because it affects whether a point is plotted as part of the function.

Piecewise Functions

Piecewise functions are functions that are defined by different expressions over separate intervals of the independent variable's domain. They allow a function to have different behaviors or rules in different parts of its domain, making them useful for modeling situations where a single rule does not apply universally.

Graphing Piecewise Functions

Graphing a piecewise function entails sketching each segment according to its respective expression and domain interval. It requires careful attention to the endpoints of each interval, as the function's value might change from one piece to another at these boundaries.

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