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

step 1 In this question we are supposed to find the domain and range of the given function which is y e

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

Key Concepts

Absolute Value Function

An absolute value function produces the absolute (non-negative) value of its input, yielding a V-shaped graph. This function is important because it introduces piecewise behavior and is frequently used as a foundation for understanding more complex constructions via transformations.

Domain

The domain of a function refers to the set of all possible input values (typically x-values) for which the function is defined. For most functions involving absolute values, there are no inherent restrictions on the inputs, unless operations like division by zero or square roots of negative numbers are present, so the domain is often all real numbers.

Range

The range of a function is the set of all possible output values (typically y-values) that the function can produce. With absolute value functions, transformations such as vertical shifts can alter the minimum value of the function's outputs. Understanding the range involves identifying how such shifts affect the lowest or highest points on the graph.

Function Transformations

Function transformations involve operations like horizontal and vertical shifts, reflections, and scalings that modify a function's graph. In the context of absolute value functions, these transformations can reposition the vertex of the V-shaped graph, directly impacting features like the range, while typically leaving the domain unchanged unless additional restrictions are involved.

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