Here are the essential concepts you must grasp in order to answer the question correctly.
Cube Root Function
The cube root function, denoted as f(x) = ∛x, is a fundamental mathematical function that returns the number which, when cubed, gives the input value x. This function is defined for all real numbers and has a characteristic S-shaped curve that passes through the origin (0,0). Understanding its basic shape and properties is essential for graphing and transforming it.
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Graph Transformations
Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the transformation applied to the cube root function f(x) = ∛x to obtain g(x) = ∛(x-2) is a horizontal shift to the right by 2 units. Recognizing how these transformations affect the graph is crucial for accurately sketching the new function.
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Horizontal Shifts
A horizontal shift occurs when a function is modified by adding or subtracting a constant from the input variable. For g(x) = ∛(x-2), the '-2' indicates a shift of the graph of f(x) = ∛x to the right by 2 units. This concept is vital for understanding how the position of the graph changes without altering its shape.
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