The transformation \( f(x) \mapsto f(x - 4) - 5 \) involves two steps that affect the graph of the function \( f(x) \):
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Horizontal Shift: The term \( f(x - 4) \) represents a horizontal shift of the graph of \( f(x) \) to the right by 4 units. This is because replacing \( x \) with \( x - 4 \) effectively moves each point on the graph 4 units to the right.
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Vertical Shift: The term \( -5 \) indicates a vertical shift downward of the graph by 5 units. After shifting the graph to the right, every point on that new graph is then moved down by 5 units.
In summary, the overall effect of the transformation \( f(x) \mapsto f(x - 4) - 5 \) is to shift the graph of \( f(x) \) 4 units to the right and then 5 units downward.
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