In Exercises 51 - 54 , complete parts ( a ) , ( b) , and ( c...

In Exercises $51 - 54 ,$ complete parts $( a ) , ($ b) $,$ and $( c )$ for the piecewise-defined function.

$$c = 2 , f ( x ) = \left\{ \begin{array} { l l } { 3 - x , } & { x < 2 } \\ { 2 , } & { x = 2 } \\ { x / 2 , } & { x > 2 } \end{array} \right.$$

Key Concepts

Continuity and Limit Analysis

Assessing continuity in piecewise functions involves examining the behavior of the function at the boundaries where the definition changes. This typically includes checking the left-hand and right-hand limits at these points and comparing them to the actual function value to determine if the function is continuous at those points.

Domain Considerations

Analyzing the domain of a piecewise function means understanding the set of input values for which each rule applies and ensuring that the function is properly defined across the entirety of its domain. This involves accounting for intervals and specific points that may have separate definitions.

Piecewise-Defined Functions

Piecewise-defined functions are functions that have different expressions or rules applying to different parts of the domain. They require careful analysis as each part of the function may have unique properties, and the overall behavior of the function is determined by the combination of these individual pieces.

Function Evaluation

Function evaluation in the context of piecewise-defined functions involves first determining which piece of the function applies based on the input value and then using the corresponding expression to compute the function’s value. This is particularly important when the input is at the boundary between different pieces.

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