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. In this case, g(x) = f(x-1) - 1 involves a horizontal shift to the right by 1 unit and a vertical shift downward by 1 unit. Understanding these transformations is crucial for accurately graphing the new function based on the original function's graph.
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Domain & Range of Transformed Functions
Horizontal Shifts
Horizontal shifts occur when the input variable of a function is altered, affecting the graph's position along the x-axis. For g(x) = f(x-1), the 'x-1' indicates a shift to the right by 1 unit. This means that every point on the graph of f(x) will move right, which is essential for determining the new graph of g.
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Vertical Shifts
Vertical shifts involve moving the graph of a function up or down along the y-axis. In the function g(x) = f(x-1) - 1, the '-1' indicates a downward shift by 1 unit. This means that after applying the horizontal shift, each point on the graph of f(x) will be lowered by 1 unit, which is necessary for accurately representing g(x).
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