Use the results of the specified exercises to determine (a) the domain and (b) the range of each

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Use the results of the specified exercises to determine (a)...

Key Concepts

Domain

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In many functions, especially those that involve the absolute value operation, every real number is acceptable, meaning the domain is typically all real numbers unless restrictions such as division by zero or square roots of negative numbers are present.

Range

The range of a function is the set of all possible output values (y-values) that the function can produce. Determining the range often involves analyzing the function's formula and any transformations applied. For functions involving absolute values, these transformations, such as vertical shifts, directly influence the minimum or maximum output values, thereby setting the lower or upper bounds of the range.

Absolute Value Function

An absolute value function is characterized by its V-shaped graph and involves the operation that returns the non-negative value of any real number. This function inherently outputs values that are either zero or positive. Understanding its properties is crucial, as it forms the basis for analyzing how further transformations (like shifts) affect the overall behavior and output of the function.

Function Transformations

Function transformations include operations such as translations (shifts), reflections, stretches, and compressions that alter the graph of a basic function without changing its fundamental structure. In the context of absolute value functions, horizontal and vertical shifts modify the position of the graph on the coordinate plane, which directly impacts the domain and range of the function. Recognizing these transformations allows for a systematic approach in determining how a base function is altered.

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