Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the changes made to the graph of a function based on modifications to its equation. Common transformations include vertical and horizontal shifts, stretches, and reflections. For example, the expression f(2(x - 1)) indicates a horizontal shift to the right by 1 unit and a horizontal compression by a factor of 2.
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Horizontal Stretch and Compression
Horizontal stretch and compression involve altering the width of the graph of a function. A factor greater than 1 compresses the graph, making it narrower, while a factor between 0 and 1 stretches it, making it wider. In the function f(2(x - 1)), the '2' compresses the graph horizontally, affecting how quickly the function values change as x varies.
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Graphing Composite Functions
Graphing composite functions involves plotting the output of one function as the input to another. In this case, f(2(x - 1)) means we first apply the transformation to x, then evaluate the function f at that transformed value. Understanding how to graph composite functions is essential for visualizing the effects of transformations on the original function's graph.
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Evaluate Composite Functions - Special Cases